The Mathematics EnthusiastVolume 6Number 4 Supplement 1 Article 1

3-2009

A Conceptual Framework for Cross-CurricularTeachingAstrid Beckmann

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The Montana Mathematics Enthusiast ISSN 1551-3440

VOL. 6, Supplement 1, March 2009

SPECIAL ISSUE ON INTERDISCIPLINARY TEACHING

FEATURED AUTHOR: ASTRID BECKMANN

Editor-in-Chief Bharath Sriraman, The University of Montana

Associate Editors:

Lyn D. English, Queensland University of Technology, Australia Claus Michelsen, University of Southern Denmark, Denmark

Brian Greer, Portland State University, USA Luis Moreno-Armella, University of Massachusetts-Dartmouth

International Editorial Advisory Board Miriam Amit, Ben-Gurion University of the Negev, Israel.

Ziya Argun, Gazi University, Turkey. Ahmet Arikan, Gazi University, Turkey.

Astrid Beckmann, University of Education, Schwäbisch Gmünd, Germany. Morten Blomhøj, Roskilde University, Denmark.

Robert Carson, Montana State University- Bozeman, USA. Mohan Chinnappan, University of Wollongong, Australia.

Constantinos Christou, University of Cyprus, Cyprus. Bettina Dahl Søndergaard, University of Aarhus, Denmark.

Helen Doerr, Syracuse University, USA. Ted Eisenberg, Ben-Gurion University of the Negev, Israel.

Paul Ernest, University of Exeter, UK. Viktor Freiman, Université de Moncton, Canada. Fulvia Furinghetti, Università di Genova, Italy

Anne Birgitte Fyhn, Universitetet i Eric Gutstein, University of Illinois-Chicago, USA.

Tromsø, Norway

Marja van den Heuvel-Panhuizen, University of Utrecht, The Netherlands. Gabriele Kaiser, University of Hamburg, Germany.

Tinne Hoff Kjeldsen, Roskilde University, Denmark. Jean-Baptiste Lagrange, IUFM-Reims, France.

Stephen Lerman, London South Bank University, UK. Frank Lester, Indiana University, USA. Richard Lesh, Indiana University, USA.

Nicholas Mousoulides, University of Cyprus, Cyprus. Swapna Mukhopadhyay, Portland State University, USA.

Norma Presmeg, Illinois State University, USA. Gudbjorg Palsdottir,Iceland University of Education, Iceland.

João Pedro da Ponte, University of Lisbon, Portugal Demetra Pitta Pantazi, University of Cyprus, Cyprus. Linda Sheffield, Northern Kentucky University, USA.

Olof Bjorg Steinthorsdottir, University of North Carolina- Chapel Hill, USA. Günter Törner, University of Duisburg-Essen, Germany.

Renuka Vithal, University of KwaZulu-Natal, South Africa. Dirk Wessels, UNISA, South Africa.

Nurit Zehavi, The Weizmann Institute of Science, Rehovot, Israel.

The Montana Mathematics Enthusiast is an eclectic internationally circulated peer reviewed journal which focuses on mathematics content, mathematics education research, innovation, interdisciplinary issues and pedagogy. The journal is published by Information Age Publishing and the electronic version is hosted jointly by IAP and the Department of Mathematical Sciences- The University of Montana, on behalf of MCTM. Articles appearing in the journal address issues related to mathematical thinking, teaching and learning at all levels. The focus includes specific mathematics content and advances in that area accessible to readers, as well as political, social and cultural issues related to mathematics education. Journal articles cover a wide spectrum of topics such as mathematics content (including advanced mathematics), educational studies related to mathematics, and reports of innovative pedagogical practices with the hope of stimulating dialogue between pre-service and practicing teachers, university educators and mathematicians. The journal is interested in research based articles as well as historical, philosophical, political, cross-cultural and systems perspectives on mathematics content, its teaching and learning. The journal also includes a monograph series on special topics of interest to the community of readers The journal is accessed from 110+ countries and its readers include students of mathematics, future and practicing teachers, mathematicians, cognitive psychologists, critical theorists, mathematics/science educators, historians and philosophers of mathematics and science as well as those who pursue mathematics recreationally. The 40 member editorial board reflects this diversity. The journal exists to create a forum for argumentative and critical positions on mathematics education, and especially welcomes articles which challenge commonly held assumptions about the nature and purpose of mathematics and mathematics education. Reactions or commentaries on previously published articles are welcomed. Manuscripts are to be submitted in electronic format to the editor in APA style. The typical time period from submission to publication is 8-11 months. Please visit the journal website at http://www.montanamath.org/TMME or http://www.math.umt.edu/TMME/ Indexing Information Australian Education Index (For Australian authors); EBSCO Products (Academic Search Complete); EDNA; Cabell’s Directory of Publishing Opportunities in Educational Curriculum and Methods Directory of Open Access Journals (DOAJ); Inter-university Centre for Educational Research (ICO) PsycINFO (the APA Index); MathDI/MathEDUC (FiZ Karlsruhe); Journals in Higher Education (JIHE); Ulrich’s Periodicals Directory; Zentralblatt MATH

Preface to special issue The Montana Mathematics Enthusiast is pleased to present a special supplemental issue this year focused on the issue of interdisciplinarity in the curriculum and classroom. This book is based on the longitudinal research carried out by Prof. Astrid Beckmann of the University of Education, Schwäbisch Gmünd- Germany, who is one of the co-founders of the Mathematics and its Connections to the Arts and Sciences international group (MACAS). In the USA, several models of connected and integrated mathematics curriculum have been implemented as well as evaluated. However, in spite of a proliferation of modeling and cross-disciplinary oriented curricula, there seems to be a lack of a specific conceptual framework for teaching. It is a misnomer to think that applied problems within the mathematics curriculum such as the solving real-life problems that deal with finding of the most efficient use of resources for a manufacturing company, or optimization problems involving scheduling and routing necessarily provide “transfer” opportunities to other real-world situations involving chemistry, scheduling of services, financial business, and ecology. It requires a concentrated effort on the part of the teacher and the learners, often involving the co-operation of teachers that teach other subjects in order to take a specific learning situation and make it truly interdisciplinary in nature. In the dominant school practices in the United States inter-disciplinarity is seen more as an exception, which is often challenged by the so-called ‘traditionalists’ who regard interdisciplinary learning as a loss of integrity and thematic curriculum as ‘squishy’ (Ferrero, 2006). Recent policy level changes in the U.S signal a shift back to traditional curricula focussing on arithmetic and algebra. In Canada, the transition in the school reform movement between the 20th and 21st

centuries was marked with more explicit emphasis on more integrative school curricula.

As my colleague, Viktor Freiman mentions, in the province of New Brunswick, the French schools go even beyond it putting a common K-12 theoretical framework for all school subjects which prioritizes the development of a new learning culture of ‘learning to learn’, making sense of learning, getting equilibrium between individual engagement and collaborative work in an interdisciplinary learning environment. Interesting results regarding interdisciplinary connections have been obtained in numerous studies conducted by members of the MACAS group such as the successful integration of literature, art and mathematics in the high school curricula ; physics and mathematics; paradoxes and mathematics; and an analysis of polymathic traits of students in interdisciplinary problem situations (see Beckmann, 2007 a, b; Beckmann & Sriraman, 2007; Sriraman & Adrian, 2004,a,b; Sriraman, 2003, 2004a,b 2005, 2007a,b). Data from projects such as the New Brunswick Laptop Initiative show how technology can be an agent of change in the classroom learning and teaching culture when interdisciplinary problem based scenarios are used to track the process of use of science, mathematics and language art to solve a real-world problems (Freiman, et al., 2007). Another experience also related to technology is a creation of a virtual interdisciplinary interactive collaborative learning community of problem solvers in math, science, and French called CASMI which only after one year of existence has attracted more than 5000 schoolchildren, university students, and teachers (Freiman, Lirette-Pitre, and Manuel, 2007).

From psychological and educational points of view there are many arguments for interdisciplinarity. Interdisciplinarity enables more connections to existing knowledge and thus leads to more complex and integrated learning. Interdiscplinarity allows more student centered lessons and increases motivation. It can also nourish reflection on specific methods of the own subject and to understand its importance. At higher educational levels, research argues for the need of so called interdisciplinary conversations that may provide a fruitful new area of exploration involving broader patterns invisible from a strictly disciplinary view (Dalke, Grobstein, & McCormack, 2006). Dalke et al., also suggest that the most generative exchange between individuals occurs as a reciprocal loop between the metaphoric relations of one individual and the metonymic structures of another, such interplay shifts perpetually generating new questions and new understandings. From its historical and cultural development, the heritage of mathematics reveals itself as a highly connected field of study. Recognition and use of connectivity among mathematical ideas, understanding how to build interconnected mathematical ideas into a whole, as well as capacity of their applications in different contexts outside of mathematics are now seen as core competence of every mathematically educated individual. In this vein, many educational systems in Canada and worldwide are implementing a more integrated and real- life connected mathematics curricula. Beyond these curricular statements, we see the utopian goal to create a humanistic approach of education, one that unifies various strands of the curricula as opposed to dividing it (Beckmann, Michelsen, Sriraman, 2005; Sriraman & Dahl, 2009). How can schooling create well-rounded individuals akin to the great thinkers of the Renaissance (Italian, Islamic)? That is, individuals who are able to pursue multiple fields of research and appreciate both the aesthetic and structural/ scientific connections between the arts and the sciences. The history of model building in science conveys epistemological awareness of domain limitations. Arts imagine possibilities, science attempts to generate models to test possibilities, mathematics serves as the tool (Sriraman, 2005). The implications for education today is to move away from the post Renaissance snobbery rampant within individual disciplines at the school and university levels. By building bridges today between disciplines, the greatest benefactors are today's gifted children, the potential innovators of tomorrow (Sriraman & Dahl, 2009). I am hopeful that the work of the MACAS group will become more and more systemic by creating new collaborations; conducting new studies; reflecting on commonalities and differences – for multidisciplinary, multicultural and divergent approaches to mathematics and its teaching and learning. Astrid Beckmann provides a useful and practical conceptual framework via which such a discourse can take place. Bharath Sriraman Editor, The Montana Mathematics Enthusiast

Missoula, Montana; February 28, 2009 References Beckmann, A. (2007a). Interdisciplinary lessons – background, arguments and possible forms of cooperation Minisymposium. In: Beiträge zum Mathematikunterricht. Hildesheim, Berlin (Franzbecker Verlag) Beckmann, A. (2007b). ScienceMath – an interdisciplinary European project. ScienceMath - ein fächerübergreifendes europäisches Projekt. In: Beiträge zum Mathematikunterricht Hildesheim, Berlin (Franzbecker Verlag).

Beckmann, C. Michelsen & B.Sriraman (Eds.) (2005). Proceedings of the 1st International Symposium on

Mathematics and its Connections to the Arts and Sciences. University of Schwaebisch Gmuend: Germany. Franzbecker Verlag.

Dalke, A., Grobstein, P., & McCormack, E. (2006). Exploring Interdisciplinairty: The Significance of Metaphoric and Metonymic Exchange. Journal of Research Practice, Vol.2, no. 2.

Ferrero (2006) Having it all: Two Chicago-area high schools demonstrate that educators don’t have to choose between innovation and traditionalism. Educational Leadership, 8-14.

Freiman, V., Manuel, D et Lirette-Pitre, N. (2007). CASMI Virtual Learning Collaborative Environment for Mathematical Enrichment. Understanding our Gifted, 20-23. Freiman, V., Lirette-Pitre, N. et Manuel, D. (2007). Building virtual learning community of problem solvers:

example of CASMI Community. Paper presented at Second International Symposium on Mathematics and its Connections to the Arts and Sciences, The University of Southern Denmark, Odense, Denmark.

Sriraman, B. (2003a). Mathematics and Literature: Synonyms, Antonyms or the Perfect Amalgam. The Australian Mathematics Teacher, 59 (4), 26-31. Sriraman, B (2003b) Can mathematical discovery fill the existential void? The use of Conjecture, Proof and Refutation in a high school classroom (feature article). Mathematics in School, 32 (2), 2-6. Sriraman, B. (2004a). Mathematics and Literature (the sequel): Imagination as a pathway to advanced mathematical ideas and philosophy. The Australian Mathematics Teacher. 60 (1), 17-23. Sriraman,B. (2004b). Re-creating the Renaissance. In M. Anaya, C. Michelsen (Eds), Proceedings of the Topic Study Group 21: Relations between mathematics and others subjects of art and science: The 10th International Congress of Mathematics Education, Copenhagen, Denmark, 14-19. Sriraman, B. (2005). Philosophy as a bridge between mathematics arts and the sciences. In A. Beckmann, C. Michelsen & B.Sriraman (Eds.), Proceedings of the 1st International Symposium on Mathematics and its Connections to the Arts and Sciences. University of Schwaebisch Gmuend: Germany. Franzbecker Verlag, 7-31. Sriraman, B., & Adrian, H. (2004a). The Pedagogical Value and the Interdisciplinary Nature of Inductive Processes in Forming Generalizations. Interchange: A Quarterly Review of Education, 35(4), 407-422. Sriraman, B., & Adrian, H. (2004b). The use of fiction as a didactic tool to examine existential problems. The Journal of Secondary Gifted Education. 15(3), 96-106. Sriraman, B., & Beckmann, A. (2007). Mathematics and Literature: Perspectives for interdisciplinary classroom

pedagogy. Proceedings of the 9th International History, Philosophy and Science Teaching Group (IHPST) 2007, Calgary, June 22-26, 2007.

Sriraman, B., & Dahl, B. ( 2009). On bringing interdisciplinary Ideas to Gifted Education. In L.V. Shavinina (Ed) . The International Handbook of Giftedness (pp. 1235-1256). Springer Science

The Montana Mathematics Enthusiast, ISSN 1551-3440, Vol. 6, Supplement 1, pp.1- 58 2009©Montana Council of Teachers of Mathematics & Information Age Publishing.

A Conceptual Framework for Cross-Curricular Teaching

Astrid Beckmann

University of Education, Schwäbisch Gmünd, Germany

Beckmann

This book is an original translation of the German original copy, 2003. Translation: Manfred Zirkel, Pädagogische Hochschule Schwäbisch Gmünd, 2008 Colloquial Translation and Editing: The Montana Mathematics Enthusiast German Original Copy: Bibliographical Information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the German National Bibliography (Deutsche Nationalbibliographie); detailed bibliographical data is available on the internet at http://dnb.ddb.de. Fächerübergreifender Unterricht Konzept und Begründung Prof. Dr. Astrid Beckmann, Pädagogische Hochschule Schwaebisch Gmuend, University of Education, 2003 ISBN 3-88120-370-2 Copy right 2003 Verlag Franzbecker, Hildesheim, Berlin

TMME, vol6, Supplement1, p. 3 Table of Contents 1 Preface 2 A Model for Cross-curricular / Subject-integrative Teaching

2.1 Initial Definition of Terms 2.2 Organization 2.2.1 Forms of Co-operation 2.2.2 Co-operative Approaches 2.3 Contacts 2.3.1 “Alien-ness” of Subjects 2.3.2 Commonality of Subjects 2.3.3 Types of Contact 2.4 Interest 2.4.1 Orientation toward Interests 2.4.2 Forms of Interest Orientation 2.5 Systemics 2.6 A Second Definition of Terms 2.7 Specializing: Applying the Model to a Particular Subject

3 The Rationale of Cross-curricular/Subject-integrative Teaching 3.1 Basics 3.1.1 Origin of Cross-curricular Teaching 3.1.2 Epistemological Bases – The Unity of the Sciences 3.2 Objectives of Cross-curricular Teaching 3.2.1 Cross-curricular Teaching as a Special Opportunity for Student Orientation 3.2.2 Cross-curricular Teaching as a Field for Holistic Learning 3.2.3 Cross-curricular Teaching as a Particular Opportunity for Motivation

3.2.4 Cross-curricular Teaching as a Field for a New Way of Thinking 3.2.5 Cross-curricular Teaching as an Opportunity to Reflect on Subject-

specific Methods 3.2.6 Cross-curricular Teaching as a “Counterpart” to Specialization 3.2.7 Cross-curricular Teaching as an (additional) Opportunity for Learning

Important Basic Mental Techniques 3.2.8 Cross-curricular Teaching as a Field in which to Experience the Social

Reality of Science 3.2.9 Cross-curricular Teaching as an Aid for the Integration and Structuring of

Learning 3.2.10 Cross-curricular Teaching as a Field for the Improved Practice of General

Competences 3.2.11 Cross-curricular Teaching as an Opportunity to Develop Ways to Deal

with Heterogeneity 3.2.12 Cross-curricular Teaching as a Contribution to General Education 3.2.13 Cross-curricular Teaching as a Special Opportunity to Deal with Topical

Issues 3.2.14 Cross-curricular Teaching as a Special Opportunity to Disclose the Impor-

tance of Interdisciplinary Co-operation in the Solution of Problems 3.2.15 Cross-curricular Teaching as an Opportunity to Solidify Subject Knowl-

edge

Beckmann

3.2.16 Cross-curricular Teaching as an Opportunity to Experience the Particular Importance of a Given Subject

3.2.17 Cross-curricular Teaching as a Special Opportunity to Tackle the Prob-lems of a Particular Subject

4 Implementation Issues 5 Final Remarks References Appendix: Overview “Cooperation Forms for Cross-Curricular Lessons”

TMME, vol6, Supplement1, p. 5 1 Preface The need for cross-curricular1 teaching has become more prevalent in recent years. There exists

a rising intensity in the recommendations of school curricula and policy documents of the fed-

eral states in Germany, such as subject-related recommendations, in frameworks of the educa-

tional policies issued by the ministries and in the general didactical debate. There are numerous

publications in which the specialized content of two or more subjects are integrated into a single

proposal for a lesson plan. The starting point of such lessons are as a rule, subject specific con-

tent matter. Any thoughts about intertwining didactic concepts are more an exception than the

rule here. So far, there has been no comprehensive theoretical backup2

1 The term cross-curricular used in this book can be taken to be synonymous with the term inter-disciplinary 2 The existing concepts of cross-curricular teaching refer at best to singular aspects (Cf.2)

.

This publication aims to fill this gap by developing a Conceptual framework/model for Cross-

Curricular / Subject- Integrative Teaching. The term model in this context is to be understood in

a pedagogical sense as a theory that analyzes didactic action on a general plane and describes it

through paradigms. Like all didactic models it has a “heuristic function” (Gudjons 1999, p.235)

in that it opens up certain problem areas and makes them the subject of discussions (i.e.: What

factors of cross-curricular teaching are enriching and which factors are problematic?). On the

other hand, it is related to “practical designing and planning, pedagogical action, concrete lay-

outing, the evaluation of and the responsibility for teaching” (Gudjon 1999, p.236). The present

model is to be understood as designed to enable teachers to make informed decisions in concrete

situations by naming the single components of cross-curricular teaching on the basis of general

didactic models as well as subject related considerations and teaching concepts.

In section 2 a model of cross-curricular / subject-integrative teaching is presented and definitions

of terms are formulated. In section 3 the theoretical framework is extended by providing ration-

ale for cross-curricular teaching based on concrete objectives.

Beckmann

2 A Model for Cross-curricular / subject-integrative Teaching 2.1 Initial Definition of Terms The following serve as initial working definitions:

First, cross-curricular teaching is a specific form of instruction. That is, we are talking about

teaching, i.e. concerning ourselves with content, a method, a way of thinking - in general with a

topic. Second, the term cross-curricular implies the existence of clearly defined subjects, disci-

plines. All pupils know about the existence of such subjects from their own experience. In chap-

ter 2 we concern ourselves with aspects of constructing a canon of subjects. Third, the term

cross-curricular implies the possibility that we go beyond a subject, i.e. that we cross discipli-

nary boundaries and in the process we touch on something else. This will by necessity be some-

thing outside the respective subject and, at least partially, something taken from different sub-

jects.

This leads us to a first general definition:

Cross-curricular teaching is instruction within a field in which subject boundaries are crossed

and other subjects are integrated into the teaching (how and for whatever purpose or objective).

On a scientific level this crossing of boundaries is made concrete through two terms:

Trans-disciplinariness is the extension of one’s own discipline to another field of work, whereas

“inter-disciplinariness is the co-operation of various related

subject areas” (Arber 1993, p.11).

This is a very important distinction, and one that is often confused in the literature.

In this context, inter-disciplinariness means the bilateral or multilateral co-operation of various

disciplines, while trans-disciplinariness means the creation of a new organizational framework

(Fried, Eizendörfer 2000, p.16). The distinction points to the fact that different forms of co-

operation and organization are possible within cross-curricular scientific work. The following

paper aims to show that this is also true with regard to school work and that cross-curricular

means and describes a very heterogeneous phenomenon.

TMME, vol6, Supplement1, p. 7 2.2 Organization

Cross-curricular teaching means the extension to other subjects or the integration of these into

one’s own subject. Therefore, successful work requires that there is a proficient co-operation

with other subjects. An individual can only achieve this through some proficiency in multiple

subjects. As a rule of thumb, this can occur though, co-operation with specialists, such as teach-

ers in the relevant subjects.

Given these considerations cross-curricular teaching implies the co-operation of various subject

teachers.

The starting points of any co-operation are the subjects or subject areas.

Alternatively, an accepted main subject.

Cross-curricular work requires initiative,

which leads to a feedback (one also talks of contributory work (IPTS).

This may be expressed by a continuous exchange.

If the initiatives come from two subjects the exchange is even more intensive.

The exchange may be so intensive that it is expressed in constant joint work.

The central topic may be subject related and may appear as a topic common to many subjects;

It may, however, also lie outside one subject and/or relate to the content of many subjects.

Topic

The lesson planning3 can be done by individual subject teachers or by the team as a whole. It may relate primarily to one subject or to various subjects at the same time. The ellipsis in the graph indicates the respective planning area.

The total planning may be characterized by intensive ex-changes.

3 Planning is here meant as a comprehensive term and refers to the continuous planning of lessons, i.e. to the reflec-tions and preparatory work as well as spontaneous decisions before or after the lessons, course or project.

Topic

Beckmann The forms of co-operation can be very different. In its simplest form, it may be restricted to sub-

ject related co-operation, enriching the individual subjects involved mutually; it may, however,

also mean joint planning of the content, the objectives and methods, the examples or courses of

instruction; and can also be expressed in joint project work.

In the following model each co-operation is composed of characteristic elements. The different

types of co-operation develop out of the combination of these elements. Each element is marked

by a symbol.

More specifically:

2.2.1 Forms of Co-operation

The various forms of co-operation result from the combination of the co-operation elements.

The rise in level goes together with an increase in the intensity and the complexity of co-

operation and the subject. For example, Levels 1 and 2 characterize the fact that the work goes

beyond one individual subject, while at levels 3 and 4 co-operation is based on the joint work of

the subjects.

In the following the various forms of co-operation are marked by their basic structures. These

basic structures can be extended through the participation of other subjects.

Cross-curricular Teaching Level 1: Topic- and Major Subject-Related Form (TM-Form)

The co-operation starts with the teacher’s realization that the teaching needs to go beyond the

subject/disciplinary boundaries and that the content and methods of other subjects need to be

used. The colleagues from other departments are involved by providing complementary aspects

that consolidate the major or central subject. The area of planning only extends to certain ele-

ments of the other subject or subjects.

TMME, vol6, Supplement1, p. 9 We speak of a special case if the teacher, who teaches in a cross-curricular way, has a multiple

competence.

Basic structure Extension (extendable)

Note:

In the present model, the term cross-curricular teaching is quite extended/flexible: it is given

level 1 standard, although the planning does not involve the co-operating subjects entirely. From

a pedagogical point of view, this still seems reasonable, as the integration of elements from other

subjects, be they physical experiments, German literature or the application of information tech-

nology, e.g. in maths instruction, always requires additional, if only minor, knowledge that ex-

ceeds the knowledge in one’s major subject (in this case: mathematics). The model is a reminder

of this – in the interest of competent instruction. (see specific discussion in Beckmann 2003a, b,

c, d).

Level 2 Parallel Topic-related Form (PT-Form)

Here, several colleagues participate in the group work from the very beginning. The starting

points are content that is common to various subjects. The teachers co-ordinate their teaching

content for the whole year aiming at teaching the common content, if at all possible, at the same

time. The emphasis of the co-operation during the school year is in particular on an intensive

exchange of ideas and temporal agreements.

Topic-related parallel work corresponds basically to the Bergheim Model. The co-operation here

is principally product-related, due to the fact that the results achieved in the parallel subjects are

to be presented in a presentation session. In the student-moderator-model the co-operation is

Beckmann extended through student papers in respective parallel subjectx. (Landesinstitut fuer Schule und

Weiterbildung, Soest 1999).

Basic Structure Extension (variable):

In special cases, one teacher takes over the co-ordination, his/her subject becomes the major subject (as a consequence, the other subjects are less represented).

Subject-integrative Teaching Level 3: Parallel Planning Form (PP-Form)

The collaboration of a team of teachers is motivated by a teaching topic that can/has to be dealt

with jointly in many subjects. Here, the teachers plan the units jointly; they are in constant con-

tact with each other exchanging ideas. It is also possible that one teacher takes over the co-

ordination and her subject then becomes the major one. The intensive exchanges during the

planning relate to the content, the methods and the objectives, the competencies to be acquired

by the students but also the implementation of the organisation (see. IPTS no year, p.7).

TMME, vol6, Supplement1, p. 11 In the planning, at least one adjournment or exchange of subjects must be taken into considera-

tion, because it may become necessary in the course of the parallel work that certain subject ar-

eas have to be finished first, before the content of another subject can be worked on.

Basic structure/ Extension (extendable): Special form: With a major subject:

Level 4: Joint Planning Form (JP – Form)

At the highest level of co-operation, the planning (as defined above) is done so closely that the

complete instruction is done in group work. Planned group work is the most complex form of

co-operation and an essential extension of the other forms of co-operation. The framework for

the instruction is a topic or topic area that can only be mastered comprehensively in collabora-

tion with several subjects. Typically, there are longer phases, in which some of the subjects in-

volved do not play a clear role. On the other hand, however, all subjects are constantly and

equally involved in the planning, so that, in principal, the special case of a co-ordination through

a major subject is irrelevant.

Due to the present spectrum of subjects, joint planning sessions exist primarily in the form of

inter-subject project work, in which the participating subjects only amalgamate for the duration

of a project. It is only rarely customary4

4 One exception is f. e. found in the natural sciences in the form co-operation between chemistry and biology or possibly physics and geography, subjects in which the various and often independent topic areas can be worked on in courses of instruction. Often, however, individual projects are also more recommendable here (Cf. Beckmann 2003b)

to orientate oneself by a course of study or a topic area

Beckmann over a whole school year, which would necessitate an extensive restructuring of the content as

well as reflections of an organizational nature (see. Gallin, Ruf 1999 for the combination Ger-

man(language)/Mathematics). As a final consequence, this form of co-operation might lead to

the dissolution of the existing spectrum of subjects and could be viewed as an ideal scenario.

Basic Structure Extension (extendable):

Subject-integrative teaching obviously goes together with a (temporary) change in the subject

structure. Huber calls this “discontinuing a subject” if instruction in the subject is discontinued

for the time of the co-operation, and “subject supplementary”, if a new integrative subject has

been created (Huber 1997). Common examples of the former are project weeks, and for the lat-

ter “supplementary instruction”, practiced at Oberstufen-Kolleg, Bielefeld or the “core courses”

at the Hutchins School at Sonama State University (Huber 1997, p.61).

2.2.2 Co-operative Approaches The form of co-operation does not specifically express anything about the time or the duration of

the co-operation. These depend, essentially, on the question of the approach chosen in the co-

operation, whether it is example-oriented, course-oriented or project-oriented. Details of each

approach follow:

Example-oriented Approach

Didactic literature offers a number of suggestions for cross-curricular teaching (e.g., see list of

examples in Beckmann 2003a). These are partly individual ideas which can be thematically

TMME, vol6, Supplement1, p. 13 classed with a group of topics or a targeted competence, which are, however, dealt with in isola-

tion and which are occasionally presented without the overall framework.

The cross-curricular idea does not refer specifically to a long-term course, and the time of its

application in school is not fixed either. Thus, such content can be integrated independently or

concurrently in the school year syllabus. They may relate to one or several subjects, or they can

integrate the students’ manifold every day experiences that cannot be tied to a single school sub-

ject. We call this approach, which refers to the teaching of a cross-curricular example, example-

oriented. Occasionally, suggestions for applications correspond to this approach.

This co-operative approach specifically concerns topic- and major subject-related work as well

as topic-related parallel work.

Course-oriented Approach

It is also conceivable to relate the cross-curricular examples to a subject-oriented topic area, re-

spectively, i.e., to plan an entire course. The term course is here to be understood as a “compre-

hensive methodological form” that serves the purpose of a “step by step instruction of a clearly

defined form of knowledge or competence” (Meyer 1994, p.143). The proposition is then

worked out on the background of the overall topic to be dealt with, and it is therefore more com-

prehensive and long-term than is the case with the example-oriented approach. We call this ap-

proach course-oriented. From a didactical point of view, it is quite imaginable on the back-

ground of this, to discard any suggestions, gained from the example-oriented co-operation, for

the overall topic. On the other hand, cross-curricular aspects may result in a restructuring of sub-

ject-specific courses.

This co-operative approach is mostly relevant for the topic-related and the structural parallel

work, but it is also conceivable in joint work.

Project-oriented Approach

Beside using the cross-curricular/subject-integrative approach in teaching an example or a whole

course, it is also possible to make a project topic the starting point of a co-operation. The co-

operation includes the typical characteristics of project work, such as (see. IPTS no date, p.8,

Ludwig 1998):

• Initiative and discussion: a project idea is presented and discussed. Pupils and teachers

participate

• Planning: Development of a project plan with all participants

Beckmann

• Execution: Independent work in small groups with regular plenary sessions

• Tests and Conclusion:

- presentation of individual work/products and summing-up of overall result

- presentation of project

- evaluation of project

Since a project topic is (if possible) situational and takes its bearings from the participants, re-

spectively, the social/practical relevance (Gudjons 1997), it is more comprehensive than an ex-

ample and, as a rule, does not relate in all aspects to a specialized course. It is generally not re-

stricted to one aspect of a particular subject but relates to cross-curricular contents. However,

project work may also be subject-specific and may come from a topic taken from the major

course involved.

Project oriented co-operation is thus not only limited to subject-integrative project work (project

orientation with regard to planning joint work) but is conceivable on all levels of co-operation.

This co-operative approach is therefore relevant for levels 1 to 4.

2.3 Contacts Each subject is characterized by certain content, methods and objectives, a special language, tool

kit and way of thinking. In cross-curricular and subject-integrative teaching these can differ-

ences clash. For instance an interpretive work from literature is quite different from the argu-

mentation of a proof. To mark these differences, the term “Fremdheit” (alien-ness), coined by

Mudroch (Mudroch 1993, p.149) in connection with interdisciplinary didactics, seems very ap-

propriate to use.

2.3.1 Alien-ness of subjects

Alien-ness of content

Subjects differ in their contents. A look at the curricula confirms this.

It is questionable, however, if this alien-ness of content plays any role at all in the context of

cross-curricular teaching. The co-operation is, after all, based on a common topic, which can

even be a topic common to one or several subjects (particularly at level 1 and 2). Here, this

TMME, vol6, Supplement1, p. 15 alien-ness does not, indeed, refer to the common overall topic but refers to the different aspects

of the content of each subject. In mathematics teaching, for example, under the topic of the cir-

cle, a particular number of points with geometric context and coinage can be dealt with, while in

physics teaching the movement of a mass point is central.

At the co-operative levels 3 and 4, the common overall topic first dissolves the alien-ness of the

content; however, in principle it is there, because the overall topic cannot, as a rule, be inte-

grated into the syllabus of a subject, and each subject also has a different approach to its content.

In addition, the unusual connection between a subject’s contents is alien, too.

Linguistic Alien-ness

Subjects differ through the language that is used in them. Linguistic alien-ness can result from

the fact that each subject has its own terms and terminology. This even applies to subjects with

similarities, for example in the natural sciences. Alien-ness may also relate to the way things are

expressed, i.e. whether a particular vocabulary is used, whether redundancy is possible, or

whether formal language is used.

Alien-ness in the objectives

The objectives of a subject are essentially formulated by its didactics. They do not only influ-

ence its general targets but also the objectives of the individual content of lessons. Didactics is

an inter-disciplinary science that has as its objective the learning and teaching of a subject in

theory and practice. In its decisions, it does not only take into account subject matters but also,

and in particular, pedagogical, psychological, social and sociological aspects, as well as educa-

tional-political, curricular, logistical and practical teaching considerations. Its objective is to

reconcile socio-political targets, anthropogenic and sociological presuppositions with the condi-

tions imposed by leisure time, and the constraints of the world of work and the environment.

Since most of the aspects mentioned are not subject related, the question arises whether there is

actually any alien-ness in the objectives of the subjects. The fields to be considered lead to the

conclusion that there is a congruence or consilience of the general objectives. It is the special

achievement of didactics to select the relevant subject areas and connect them to educational

objectives in such a way that they can contribute to their realization. This may, on the other

hand, result in differences between the objectives of the individual subjects, or it may lead to a

difference in the emphases, put on certain objectives. It may, for example, be a social objective

to familiarize students with the methods of the natural sciences. Physics, chemistry or biology

Beckmann are more likely to adopt such objectives than the arts. In spite of a general agreement amongst

subjects, as regards their objectives, there is alien-ness in details.

Alien-ness in Ways of Thinking

When students in a school project, conducted by Schneiter and Zimmermann, found out that

their definitions, formulated in maths, were shown to the German teacher, they were enraged:

“They felt deceived; said that it was a mean deceit, that they had paid no attention to the linguis-

tic form and that they felt exposed now5

The alien-ness in the various subjects’ ways of thinking is especially obvious in physics instruc-

tion. One of the objectives there is to familiarize students with the methodology of physics and

its way of thinking. What is special about this approach is the concept of nature in physics.

Physics means “the reduction of man to a physicist” and “the reduction of nature to an object of

physics” (Oy 1997, p.29). What is meant here is, that “man as a physicist can do without certain

senses and perceptions. He does not, for example, need a sense of temperature and the ability to

distinguish between colours.” Additionally, it means “that certain natural phenomena cannot be

registered by the physicist” or, that among the natural qualities registered, “their special traits

such as the red of the colour red or the quality of a sound, are not of importance to the physi-

cist.”

” (Schneiter, Zimmermann 1985, p.48). This story high-

lights, beside the above mentioned linguistic differences, the alien-ness in the way of thinking,

that shows in different subjects. While in ones native tongue, literature teaching, is an open way

of thinking, including verbal imperfections, is acceptable, mathematical thinking is target-

oriented and is, even during the search for a solution, presented in such a way as if it is only ad-

dressed to a mathematically informed audience (see Gallin, Ruf 1993, Wagenstein 1982). Thus

one suspects that these traditional ways of thinking, in spite of recent publications on maths di-

dactics, have not been replaced everywhere.

6

Different subjects differ in their methods. These differences were already pointed out above with

regard to language of methodology. In general, the methodology used by different subjects, is

described by their respective didactics. They are closely connected to their objectives, and as far

(Oy 1997, p.29)

Methodological Alien-ness

5 Editorial note: The author is alluding to the fact that students often do not pay explicit attention to the rules of grammar, syntax and semantics when writing mathematics, given their prevalent notion that mathematics is more about the notation and formalism, as opposed to description and exposition. 6 sic

TMME, vol6, Supplement1, p. 17 as they relate to general objectives and the didactic principles attached to them, they are quite

similar.

However, methodological alien-ness is one of the most conspicuous characteristics when we

compare school subjects. This may relate to the individual methods learned and applied, such as

writing a satire in a German language class, doing sketches in an art class, applying the Pythago-

ras theorem in mathematics, writing the notes of a short tune in music, planning an experiment

in chemistry and dribbling in sports; however, it may also refer to the entire course structure,

such as the inductive/deductive7

2.3.2 Commonalities of Subjects

method in physics.

Since different subjects are characterized by alien-ness, the question is, aren’t there any similari-

ties? Possible similarities have already been mentioned in the analysis above. They relate in par-

ticular to the overall objectives and also to individual topics. There are certain parallels among

“related” subjects in the language and the methodology used, for example grammar in languages

and the inductive/deductive method in the natural sciences. Common features may be reasons

for cross-curricular teaching. Co-operation is, however, only interesting here, as long as each

subject has its own characteristics as regards content, concept and methodology. Cross-

curricular/subject integrative teaching lives off alien-ness.

The alien-ness of subjects does not make a statement about its instructional implementation.

This leads us to an extension of our co-operation model in that the contacts with “alien” aspects

(alien content, methods, ways of thinking, objectives, language) are taken into account and dif-

ferent forms are distinguished.

2.3.3 Types of Contact

In co-operating, the teachers learn the alien aspects of other subjects through feedback (TM –

Form), through agreements (PT-Form), through intensive exchanges (PP-Form) or joint work

(JP-Form). The instructional implementation, however, requires that the teachers involved actu-

7 Editorial Note: Inductive/deductive in the sense of Francis Bacon (1561-1626)a controversial figure in the history of the sciences. Bacon’s rejection of the Aristotelian tradition, and formulation of the inductive method for science should rightly place him as one of the fathers of modern science. Newton's Principia and Opticks reflect the use of inductive arguments as a methodological principle carefully spelled out by Bacon in the Novum Organum nearly sixty years earlier.

Beckmann ally realize the alien aspects and are capable of demarcating them from aspects of their own sub-

jects.

This requires a fixed terminology:

On the Term Idiosyncratic Aspects (IA)

Starting from a certain subject, the term idiosyncratic aspect denotes the subject areas (content,

methods, objectives etc.) which are validated by the subject’s didactic principles. These may be

general aspects, such as the deductive method, but also singular aspects like doing classical

geometrical constructions with straightedge and compass.

On the Term Alien Aspects (AA)

Starting again from a particular subject, the term alien aspect refers to the idiosyncratic aspects

of a different subject (co-operating subject), which differ from the aspects of the original sub-

ject.

The differences may be total, e.g. German literature as subject material, or may only concern

parts of a common aspect. Alien aspects can thus be recognized by comparing common aspects.

In a co-operation, the idiosyncratic aspects of the individual subjects become alien aspects as far

as they distinguish themselves from those of the co-operating subject.

The contact with alien aspects can be closer or less so and is expressed in three ways:

Use of Alien Aspects

The teacher can use the alien aspects to enrich the topic through the use of alien contents or ob-

jectives, or master the topic by using different methods, a different language or a different way

of thinking.

One example of this is given in the support given to mastering a mathematical topic by using the

methods and contents of a German language class, such as individual text production (Maier

2000, Selter 1994), or the use of German literature (Beckmann 1994, 1995, 1999). Another ex-

ample is the use of experiments (physics methodology) in maths teaching (Beckmann 1999a).

The use of alien aspects is most evident if the teacher was made familiar with these through

feedback, respectively supportive work.

TMME, vol6, Supplement1, p. 19

Integration of Alien Aspects The teacher can connect alien aspects with subject specific aspects, respectively teach them in

connection with aspects of his/her own subject. This can be done by relating them to the instruc-

tion running in parallel or directly. Directly here means, that not only the subject aspect is

broached or applied in class or in a joint project but also the alien aspect.

Examples of this are broaching the subject of common forms of presentation in chemistry and

maths (Köhler 1992), the aesthetic reception or interpretation of a literary work containing a

maths problem (Beckmann 2003c), the description of a tune through vectors in music (Wilke

1976), or the repeated application of algorithms to mathematical objects for the creation of

works of art such as the Sierpinski Gasket (Peitgen, Jürgens, Saupe 1992).

Possibilities for integration ensue in particular, whenever alien aspects clash during the ex-

change of ideas, when the teaching procedure was agreed upon beforehand and the integration

reflected upon jointly.

The more intensive the exchange of ideas, the more intensively the integration can be realized.

The results in differences with respective alien aspect to become blurred.

Mixing Aspects

In planning a subject integrative teaching sequence or a joint project, a particularly intensive

discussion about the alien aspects takes place. The alien aspects do not just clash, but they have

to be put in relation to each other, which can go beyond simple integration, or is at least a form

of particularly comprehensive integration. Their connection must make it possible to carry out a

joint teaching sequence or a joint project. As a last consequence, this could lead to partially re-

linquishing the system of one’s own subject and to a re-orientation.

Beckmann 2.4 Interest Cross-curricular/subject-integrative teaching can potentially be meaningless, unless the contact

goes together with an enrichment (in a positive sense) of the subjects involved (Mudroch 1993).

Thus, the interest in cross-curricular/subject-integrative teaching must always be directed at en-

richment (to avoid any misunderstanding: We are not talking about the personal interests of

teachers, but the pedagogical-didactical interest directed at the students for the sake of success-

ful instruction), in order to preserve the intellectual integrity of the subjects being taught.

An enrichment of the subjects can be achieved by considering their alien-ness, or (potentially)

their common features.

It is therefore imaginable that the scientific method, common throughout the natural sciences,

can be used more comprehensively in more subjects. What is more, in the alien-ness of the sub-

jects there seems to be a special opportunity for cross-curricular teaching. On the one hand, it

makes it possible to deal with a multitude of topics, unrelated to the subject itself, and on the

other hand, it opens up new approaches to subject-specific contents. This way, different types of

interest can be distinguished.

2.4.1 Orientation towards Interests

Co-operation may be motivated by a singular interest but may also involve several interests at

the same time; it may orient itself to alien-ness or (potentially) to common ground

.

• Orientation towards content: The inter-subject approach facilitates the work on a particu-

lar content.

For example, certain biological questions, such as research into the biological population

of a pond, can only be answered with the help of mathematics (Cf. Shahani, Parsons,

Meacock 1979).

• Orientation towards Methods: The inter-subject approach facilitates the working out of a

particular method (including linguistic method).

TMME, vol6, Supplement1, p. 21

For example, the teaching of mathematical transformations in the context of musical

transformations (retrogression) can be supplemented by the musical effects achieved

through it (Dienes 1987, Nisbet 1991).

• Orientation towards Competence: The inter-subject approach promotes certain compe-

tences. In cognitive psychology, competences are individual prerequisites for behaviour

and types of experiences (Flammer 1999, p.19).

For example, carrying out and evaluating a physics experiment can activate functional

thinking through the need for switching between different forms of presentation

(tables, graphs, functional equations etc.).

• Orientation towards Ways of Thinking: The inter-subject approach improves the access

to certain ways of thinking.

2.4.2 Forms of Interest Orientation

Interest orientations may take various forms.

Subject Orientation

The starting point here are two (or more) subjects (S1, S2 …) aiming at a co-operation.

S1 is used here to introduce, discover, work out, explain, describe connections with S2 in order

to create or structure objects in S2. On the surface, it is all about teaching something about S2.

That way, using S1 is an enrichment of S2. On the other hand, S1 also profits from this. By us-

ing S1 to teach S2, its importance for topics outside that subject becomes obvious.

This form particularly based on the alien-ness of the subjects. As the relation to the subject is

dominant, it mainly concerns cross-curricular teaching and subject-integrative teaching less so.

Parallel Orientation

The characteristic of this form is the parallelism of different subjects S1, S2 … . We can distin-

guish two types, depending on how closely the common features are treated:

I) The subjects are worked on in parallel in that

- a common aspect (topic, method etc.) is worked on jointly

Beckmann

- common aspects of the subjects are worked on simultaneously, on the basis of a sub-

ject-related topic (e.g. from S1).

An example for this is working on the parable in German and maths (Beckmann 2000) or the

circle in physics and maths (Beckmann 2003b).

II) The class reflects, consciously and simultaneously, on the common features of both subjects

by making the common aspects their topic in both the subjects involved.

One example of this is consciously making these formal aspects the topics of chemistry and

maths instruction, however, with different intentions (Köhler 1992).

Indicators of this form are the (seemingly) common features of different subjects. The common

topics or the common aspects (methods, ways of thinking) open up the opportunity to work on

these related aspects in a more varied but also more critical way.

Comprehensive Orientation

The subjects can also be enriched in the way that each subject makes a contribution to the work

on a topic that is not tied to a particular subject. The subject integrative topic forms the starting

point of a co-operation. It has to enable all subjects involved to make an informed contribution.

The characteristics of this approach is

- the use of subject-related aspects in non-subject contexts

- the experience of the importance of subject-related aspects for non-subject contexts

- the combination of subject and alien aspects.

In comprehensive orientation, alien-ness plays an important role, in that each subject has its own

special access to the topic. On the one hand, it contains a special potential for comprehensive

work, on the other hand, however, there is a problem. In comprehensive orientation the forms of

alien-ness crash more directly than in subject orientation, where the alien aspect is used con-

sciously for enrichment, or in parallel orientation, where the discovery of alien-ness is the actual

support of the co-operation. In a comprehensive orientation approach to teaching, alien aspects

might converge in such a way, that they exclude each other in the worst possible case. Co-

TMME, vol6, Supplement1, p. 23 operation in comprehensive orientation might therefore require a change in the systemic of a

subject.

Such co-operation requires the teachers to have sound knowledge of all aspects of their subjects.

First of all, the subject teacher must be able to recognize that a topic is relevant for his subject

(roughly corresponding to the magnet mode in projects, Ludwig 1998). Secondly, s/he must be

able to correlate the subject’s aspects to the topic (roughly corresponding to the star mode, here,

however, applied to any subject and related to those aspects, Ludwig 1998, cf. graph).

Due to the fact that the starting point is a non-subject topic, this characteristic mostly concerns

subject-integrative forms of co-operation.

Explanation: Comprehensive orientation requires a topic that concerns (attracts) one’s own sub-

ject, and that one has a complete survey of which subject-related aspects the topic contacts.

2.5 Systemics Scientific research takes place within a systematic canon or system of thought, determined by

the respective discipline. With regard to school subjects, the didactics of a subject determines

the systematic framework for instruction. It determines the questions, methods and objectives.

Each subject is thus given its own set of systemics.

Cross- curricular teaching is characterized by co-operation, in which contact is made with alien

aspects. Mastering a topic requires coming to terms with the alien aspects of the other subjects.

The question is, what kind of consequences does the co-operation of different subjects have for

the subject-specific systemics, e. g., does cross-curricular teaching have to lead to (temporarily)

giving up the systemics of the individual subject, respectively does cross-curricular teaching

follow its own systemic?

Beckmann The question whether, or in how far, the systemics of the subject deviates from the systemics

prescribed by its didactics, obviously depends on the level of co-operation, its starting point, the

form of contact and the interest orientation. In the following table, the possibilities of cross-

curricular/subject-integrative teaching are collected. It has to be observed that for certain forms

of co-operation, due to their structure as described, certain forms of contact must be excluded. In

the same way, example-oriented co-operation does not take place at all levels. Theoretically, it

is thinkable to practice subject-integrative co-operation (level 3 and 4) in an example-oriented

way. However, as the co-operation is very tight here and is based on planned work, the practical

example will rather turn into a (small) project.

Le-vel

Form of Co-operation

Co-operative approach

Form of Contact

Interest Ori-entation

Characteristic

1 Cross-curricular

TM – Form Topic and Major subject -related-Form

example-, course-, project-oriented

use of Alien As-pects (if nec. integra-tion)

alien-, com-munal-oriented (by Alien As-pects, by Idiosyncratic Aspects) content-, method-, competence-, thought-Oriented

subject-parallel-Oriented

2 Cross-curricular

PT – Form Parallel Topic –related - Form

example-, course-, project- oriented

use and integration of Alien Aspects

subject-, paral-lel- orientation (resp. compre-hensive orienta-tion)

3 Subject-integra-tive

PP – Form Parallel Plan-ning – Form

course-, project- oriented

use, integration, mixing of Alien Aspects

subject-, paral-lel-, comprehen-sive orientation

4 Subject- integra-tive

JP – Form Joint Planning Form

course-, project- oriented

integration of Alien Aspects, mixing Aspects

comprehensive (also parallel orientation)

Table 1 Possibilities of cross-curricular/subject-integrative teaching The table above indicates that a rise in the levels is connected to a decrease in subject systemics

towards an original subject-integrative/cross-curricular systemics. As long as the topic- and sub-

ject-related work only makes use of the alien aspects, the major subject can essentially retain its

autonomy and thereby its own subject didactics. Co-operation at the higher levels, however, in

particular the integration of alien aspects, mixing them with alien aspects, but also project orien-

tation, may mean a tighter connection, which makes keeping the systemics of a subject at least

somewhat more difficult.

2.6 A Second Definition of Terms The theoretical considerations lead us to an extension of the first definition: Cross-curricular/subject-integrative instruction means dealing with a (subject-related or non-related) topic, in which the subject borders are exceeded and other subjects are inte-grated.

TMME, vol6, Supplement1, p. 25 The instruction is done in co-operation. The co-operation can consist of topic- and major subject-related work (TM-Form, level 1), of parallel topic related work (PT-Form, level 2), of parallel planning work (PP-Form, level 3) and of the planning of joint work (JP-Form, level 4). It may be example oriented, course oriented or project oriented. Instruction on levels 1 and 2 is termed cross-curricular, on levels 3 and 4 it is termed sub-ject- integrative. Exceeding subject borders results in contact with other subjects. Here, common interest (Idiosyncratic Aspects) but also alien interests (Alien-ness) meet. They may be related by making use of the alien aspects, by integrating the alien aspects or by mixing subject and alien aspects. The interest in cross-curricular/subject-integrative teaching lies in an enrichment of the sub-jects. Here, the interest can orient itself to common aspects or to alien aspects. It may be oriented to content, methods, competences and ways of thinking, and it may appear in the form of subject orientation, parallel orientation and comprehensive orientation.

2.7 Specializing: Applying the Model to a particular Subject The model for cross-curricular teaching describes the theory of any cross-curricular instruction, i.e. the theory of cross-curricular maths, German, physical education etc. Cross-curricular/subject-integrative maths instruction (German, physical education etc.) is a form of instruction, in which one of the subjects involved is maths (German, physical edu-cation etc.).

In relation to a particular subject X, there are certain specializations (X stands for a particular

subject and is replaced by M for maths, D for German, S for phys. ed. etc.):

In topic- and major subject-related work, subject X is the major subject.

In the special forms of topic related work and parallel planning, subject X can be the major sub-

ject. In this case, the co-operation orientates itself to the interests of subject X. The alien aspects

are on principle taken from other subjects. These are used, integrated or mixed to deepen the

content, the methods etc. of X. Two kinds of subject orientations derive from this.

X-orientation (M-orientation, G-orientation, S-orientation etc.)

The starting point here is subject X. In the co-operation with subject S the content, methods etc.

of X are applied (and possibly worked out) to introduce correlations in S, to discover, work out,

explain, describe, create objects in S or to structure S.

Beckmann We call this form of subject orientation X-oriented, since subject X plays a decisive role here.

On the surface, the objective is to teach content from S. However, this content is developed and

derived with the help of X. The basic approach is to teach the meaning of X (e. g. maths) by

showing that X is necessary for the development of a certain content outside X. From the point

of view of X this means, that the topic to be dealt with, is embedded in a content, taken from F,

which is developed with the help of these connections from X.

S-orientation

The starting point here is the related subject S. Subject S is used to introduce connections in X,

to discover, work out, explain, describe them, create objects in X or to structure X.

We call this form S-oriented, since the subject S is of particular importance here. On the surface,

it is about the teaching of contents, methods, taken from X; these are, however, only absorbed,

deepened or taught in an enriched way through the contents of F.

3 The Rationale for Cross-curricular / Subject-integrative

Teaching8

3.1 Basics 3.1.1 Origin of Cross-curricular Teaching The roots of cross-curricular teaching can be found in educational progressivism, which devel-

oped at the turn of the 19th/20th century (up to about 1933). Many of the objectives of cross-

curricular teaching correspond to the principles of educational progressivism (see section 3.2).

Educational progressivism resulted from the need to reorganize school for more humane learn-

ing. The “humane school”9

8 For the sake of clarity, we will only speak of cross-curricular teaching in this chapter but subject-integrative teach-ing is meant here as well.

was to be based on the freedom of the child (Montessori), his/her

9 Editorial note: The historic evolution of didactic traditions in Germany, France and Italy reveals several commonalties which can serve as a fertile source of discussion and for mediating common agendaz. In all three countries mathematics didactics has its origins in the humanistic movement that emerged in post Renaissance Europe. Humanism is typically viewed as a literary and cultural movement which began its spread through Western Europe in the 14th and 15th centuries. However histories of the Renaissanceindicate that humanism was also an educational curriculum with its roots in 14th century Italy with Guarino Veronese (1374-1406) in Ferrara and Vittorino da Feltre (1373-1446) at Mantua who began using Quintilian texts as a model for their educational program. The Italian poet and lawyer, Francesco Petrarca, or Petrarch (1304-74), is called the “father of humanism”. The widespread transmission of classical texts with the serendipitous advent of the printing press led the humanistic movement into Germany, France and England where it mutated differently. In France, as we previously described, one of the mutations of humanism was the encyclopaedic tradition, which is characterized by

TMME, vol6, Supplement1, p. 27 needs (Freinet) and on the laws of human development (Steiner) with due consideration of soci-

ety’s needs. The humane school therefore had to be a “new” school, a school that became the

lifestyle of the students (Schoning 1999, p.22). The special characteristics of this school are the

observation of knowledge derived from developmental psychology, the reference to real life

(Freinet), free working time (Petersen, Montessori), the combination of practical and intellectual

work (Kerschensteiner), holistic learning (Leipziger Lehrerverein etc.). In connection with

cross-curricular teaching, Bertold Otto (1859 – 1933), who created a form of collective class,

has to be mentioned in particular: at least three times a week instruction was to take place

jointly, independent of the canon of subjects and the individual syllabuses, and it was to orient

itself to the wishes of the students (Gudjon 1999). According to the model of cross-curricular

teaching, this corresponds roughly to subject- integrative teaching on level 4.

Even if the idea of cross-curricular teaching was made concrete 100 years ago, this does not

mean that it was generally accepted. School is, and has always been, conditioned by so-

cial/societal factors. A look at educational history points up the constant

“struggle to liberalize and democratize educational opportunities in the public school

system”, which is a multi-faceted, controversial process, which has been influenced by

the political and economic needs to modernize, which again and again came up against

barriers … which - once set in motion – can never be stopped completely”. (Herrlitz,

Hopf, Titze 1998, p.9).

Curricular decisions have to be seen on this background. Thus, Bertolt Otto can also not be

called the inventor of cross-curricular teaching but someone, who cumulated ideas developed

before him. The general basic ideas can be found in Rousseau (1712-1778), who pleaded for the

child’s original contact with nature, in the philanthropists (a pedagogical movement from around

1750 to 1800), who postulated an “alternative to Latin scholarship” and an “utilitarian education

with the objective to prepare children for usefulness in an economic and social environment”

(Gujons 1999, p.89) and surely in Pestalozzi (1746 – 1827), who saw demonstrability and ex-

perience as the basis for learning.

the principles of universality, rationality and utility In Germany, humanism in its new forms, mutated into a child-centered epistemology (or an emphasis on the individual), which have influenced the educational traditions of Anglo-Saxon countries (Sriraman & Törner, 2008). Reference Sriraman, B & Törner, G. (2008) Political Union/ Mathematical Education Disunion: Building Bridges in European Didactic Traditions. In L. English (Ed) The Handbook of International Research in Mathematics Education (2nd Edition). Routledge, Taylor & Francis, New York. pp. 660-694.

Beckmann

Cross-curricular ideas can also be found in Humboldt’s idea of a general education with the

three elements (Gudjons 1999, p.94):

• individuality (one’s own subjectivity)

• totality (forming all powers to wholeness instead of filling up students with material)

• universality (practical and intellectual education)

In cross-curricular teaching, works by Herbart (1776 – 1841) have to be considered. Herbart

developed a basic concept of cognitive psychology with the elements of absorbing, thinking,

assimilating and applying. This classification describes the concentration and the structure of a

“circle of ideas” and supplies the formal basis for lesson planning, which Herbart was later criti-

cized for by the reform pedagogues. His disciples tried to develop this into a system of culture

levels. It is interesting to note that this already contained typical cross-curricular ideas:

“With their idea of concentrating teaching they wanted to avoid overcharging the cur-

riculum through too many subjects, an attempt that led to some absurd concentrations:

…from the sea that Columbus crossed to the characteristics of water (physics and chem-

istry) and from their to Schiller’s poem The Diver (German literature).” (Blankertz in

Gudjons 1999, p.101; ann: The idea, Blankertz calls “absurd”, might well be a project

idea in modern subject integrative teaching).

The interest in educational progressivism today and its clear influence is not at all surprising.

Today’s basic pedagogical questions are similar to those 100 years ago, even if the historical

background is different.

“The historical parallel is amazing, though: Praxis gives new impulses, theory attempts

to clarify and systematize it afterwards.” (Gudjons 1999, p.102).

3.1.2 Epistemological Bases – The Unity of the Sciences From an epistemological point of view, “the idea of the unity of science” serves as the back-

ground to cross-curricular teaching. It means the following: (Mittelsträß in Gräfrath, Huber,

Uhlemann 1991, p.1):

The unity of science exists or a well founded programme of its representation,

1. because the world as the object of science is one (unity of nature)

TMME, vol6, Supplement1, p. 29 2. because the scientific interest is one (it is the rational understanding of the world)

3. because the criteria of rationality, and by implication the methods also, that guide sci-

entific work, tentatively refer to unity (criteria for rationality are expressions of the same idea of

scientific rationality)

4. because, philosophically, the idea of reason is one and is the highest criterion for ra-

tional action

5. because the researching subject is one and science is not without a subject.

The idea of the unity of science is not at all new. In antiquity, this unity resulted by itself as the

ontological unity of nature. In modern times, this view changed in that there is no objective

unity of science but that science is seen as an idea of human reason (Gräfrath, Huber, Uhlemann

1991). Two typical concepts of unity, which are based on the unity of human reason but which

are still as concepts essentially different, are those of Leibniz and Kant. Kant’s normative-

transcendental approach is based on the idea that knowledge is not given a priori by primary

cognition and action but that people have to discuss their experiences and through this establish

meaning. Leibniz’ holistic approach on the other hand, is based on the logical structure of the

world, made up of inseparable units and a fixed method of their combinations.

Another modern philosophy that, in the 20th

As a consequence of this guiding principle, the idea of cross-curricular instruction plays an im-

portant role in logical empiricism. The motive behind the search for the unity of science was the

idea that work on complex questions, necessitating the transgression of individual subjects, re-

quires unifying respectively relating the terms and theorems of the diverse disciplines. On the

other hand, the verification of a theorem requires the unification of the language used by the

theorist who puts forward the hypothesis, as well as that of the observer who examines the hy-

pothesis. However, in the implementation of these ideas logical empiricism reached its limits,

resulting in a series of suggestions for improvement, such as the falsification method by Popper,

i.e. the verification of negations as a criterion. All in all, one can state that the question of the

century, pursued the idea of the unity of science is

the logical empiricism of the Viennese circle. Their main aim was to bring together the knowl-

edge, gained in the individual sciences, in a unified science. Their basic theorem is: All scientific

knowledge is grounded in experience. The idea that statements about experiences need to be

empirically controlled, results in the theorem on meaning: A statement is meaningful if it can be

verified.

Beckmann unified concept of logical empiricism could not be solved satisfactorily. It is remarkable on the

other hand, that certain applications of the concept even led to devisive scientific theories (e.g.

Comte’s division of the experimental disciplines).

A trend that has established itself since about 1940 and that pursued the idea of the unity of sci-

ence, has become known by the name system theory. Important representatives are Bertalanffy,

Rapaport, Boulding as well as Wiener. From a theoretical point of view, the system theory ap-

proaches correspond to the holistic approach. The characteristic feature here, however, is the

importance of the self-reference, self-regulation of elements, respectively the mutual effects of

the different elements of a unit. In this context, it is worth mentioning Piaget, who drew special

conclusions from system theory for the trans- and inter-disciplinary qualities of subjects (subject

combinations at science level) (Vonèche 1993): Piaget’s basic term is structure as a system of

existing transformations whose results are again part of the system. It is Piaget’s idea that this

mechanism characterizes every scientific discipline. Depending on the kind of relationship the

special sciences have with each other, there will be progress. In multi- or pluri-disciplinary

situations an exchange of information takes place that does not, however, lead to self-regulation.

The interdisciplinary character implies activities between the sciences. These are, according to

Piaget, compulsory and the differences remain. Piaget therefore pleads for the over-arching dis-

cipline, trans-disciplinariness, where all elements (here, the special sciences with all their fac-

tors) interact. Only this way, self-regulation is possible. The interaction between maths and

physics (Cf. e.g. Zeidler 1998) is a good example of the progress made possible through trans-

disciplinariness.

“Maths enlivens physics. However, once activated, maths receives repercussions from

physics that change it… This dual movement towards a renewal is in itself a sign for the

move in the sciences” (Vonèche 1994 in translation, p.115).

Transferring these thoughts to cross-curricular teaching would mean that there is only a gain in

the mixture of alien aspects and renewal (Cf. model). However, Piaget’s point of view is quite

extreme here (Cf. complementariness). In addition to this, cross-curricular teaching does not

only serve the purpose of progressing knowledge but pursues a series of other goals

(Cf. 3.2).

More recent theories of science are more modest with regard to the idea of the unity of sciences.

They are only discussing individual criteria as the measure of their characterization, such as a

TMME, vol6, Supplement1, p. 31 uniform scientific terminology, uniform scientific laws, methods or contents (Gräfrath, Huber,

Uhlemann 1991).

A transfer of the idea of the unity of science to schools can be of importance for cross-curricular

teaching, in that it, e.g. draws attention to the possibility of difficulties in understanding between

different subjects. In addition it motivates people in their search for the unity of all school sub-

jects. Indeed, looking at uniformity from a superior point of view one can observe that the gen-

eral objectives of all subject didactics are similar, in that they take into account concepts of de-

velopmental psychology and social concerns in teaching suggestions. A certain uniformity of the

subjects shows in their respective teaching concepts. Examples are independent study and inter-

active instruction.

However, the unity of all subjects does actually not correspond to the interests of cross-

curricular teaching. According to the model of cross-curricular teaching, the teaching across

subjects only becomes interesting through alien-ness. From a theoretical scientific point of view,

this corresponds exactly to its meaning: The idea of the unity of science is not in opposition to

multi-disciplinary teaching. It is simply rather “unity in variety”. The necessary specializations

in the different disciplines are accepted, however, the disciplines and their knowledge are not

seen in isolation but within an interconnected system. For its description the term complemen-

tariness is well suited (Gräfrath, Huber, Uhlemann 1991, p.3).

“Each discipline has its own access to the world, but points beyond itself, insofar as it

cannot grasp the world (scientifically) by itself. Problems of science don’t adapt to the

borders drawn up by the historically grown disciplines. They often make it necessary that

a scientific discipline and the individual scientist keep an eye open for cross-curricular

questions. Transgressing ideas of unity find, on the one hand, a methodological expres-

sion in the application of rational criteria and on the other hand become obvious in their

understanding of science as unified research, in which the drawing of borders by individ-

ual disciplines is second rate.”

3.2 Objective of Cross-curricular Teaching The objective of cross-curricular teaching has already been integrated into its model. It can be

found there under the dimension of interest :

Beckmann The objective of cross-curricular teaching must be enrichment. Here, the interest can orientate

itself to the common features or to alien-ness. It can be orientated towards content, method,

competence and way of thinking, as well as to the characteristic forms of subject-, parallel- and

comprehensive-orientation.

The decision to teach in a cross-curricular way must therefore be preceded by an analysis of its

objective, which ensures (to a large extent) that enrichment is part of it. In detail, this requires

e.g. the consideration that acquiring a subject competence (with competence and subject-

orientation) is made easier through alien aspects. This requires, however, a basis for such

judgement, such as higher ranking criteria, that are applicable to the respective individual case.

At the same time, the higher ranking criteria form a rationale for cross-curricular teaching and

arise, apart from subject aspects, from different disciplines such as education, psychology and

the theory of science.

We will now present the objectives of cross-curricular teaching and explain them briefly. It will

be shown that the arguments speak in favour of the fact that these objectives can be reached in

cross-curricular teaching (which, for the moment, excludes any statement about subject-specific

teaching). On the background of a concrete implementation, they may, however, appear to be

quite general, for, according to the cross-curricular model, there is not just one type of cross-

curricular teaching but quite different ones. In addition, not every objective is prevalent in every

grade:

In the first grades in secondary I, (as well as in primary school) those objectives that en-

able students to have concrete and graphic experiences (in particular objectives 3.2.1 to

3.2.3, 3.2.7, 3.2.9 to 3.2.12) are predominant.

In the middle grades of secondary I, complex questions and general principles, the refer-

ence to reality with its social, technological etc. developments become more and more

important (objectives 3.2.1 to 3.2.3, 3.2.5, 3.2.7, 3.2.9 to 3.2.15).

At the end of secondary I and in secondary II a new dimension is added, notably (con-

scious) reflection and making cross-curricular work the topic of the work in class (in par-

ticular objectives 3.2.4 to 3.2.8, 3.2.14, 3.2.15, but all others as well).

The decisive question, however, is whether the advantages connected with the objectives are

actually real ones (expectably so). Cross-curricular teaching requires a critical discussion of the

objectives in each individual case – and in comparison with subject-specific teaching.

TMME, vol6, Supplement1, p. 33

In the following, possible objectives are listed.

3.2.1 Cross-curricular Teaching as a Special Opportunity for Student Orien-

tation

Student orientation is a comprehensive term referring to the general approach in educational

progressivism “student oriented learning”. It is based on results from the psychology of learning

and developmental psychology, as well as on a concern for the students’ perspectives and inter-

ests. As a pedagogical principle, student orientation has grown out of theories that look at the

student as a person and as a cognitive but also affective and social being.

A series of psychological theories is available for the observation of developmental-

psychological aspects. From the teacher’s or the pedagogue’s point of view, it is of interest to

know, which conditions have to be created to have a positive influence on learning and devel-

opment. Learning means the achievement of small, short term and often temporary changes in

competence, habits, values, physical strength etc. . The modern view of learning starts from the

assumption that structures (concepts, rules etc.) are innate in man and in the child; that these

have to be related to the experiences (stimuli) made in the world, checked and, if necessary,

modified accordingly. Learning means changing these structures: learning is structural (Cf.

Lefrancois 1994, Flammer 1999).

The central notions of modern cognitive psychology are attention and memory (inclusive of re-

membering). Learning means, paying attention. Only those stimuli, that get attention, are se-

lected, processed, coded, i.e. formed terminologically. Attention is influenced by structures al-

ready in existence. Despite the fact that coded information is stored for life (long term memory),

it cannot always be called up. The ability to call up information, respectively remember things,

is not only important in itself, but also for the transfer of knowledge. Certain stimuli, either ex-

ternal or provided by the learner on account of existing information, are of crucial importance

for our memory. Here, strategies for organizing material, thus the abstraction of meaning from

material (Lefrancois 1994, p.175), are helpful besides repetition and structuring.

Development is characterized by long term changes (in most cases, of several areas simultane-

ously) and includes processes conditioned by maturity. Development is also structural, i.e. not

Beckmann only single elements change, such as abilities, opinions etc., but also the structure of these ele-

ments, the total structure. Flammer marks development as the class of those changes that incur

further changes (Flammer 1999, p.19). There exists a mutual effect between learning and devel-

opment: learning processes can trigger off developmental processes, development makes new

learning processes possible. The developmental theories name various activators for develop-

ments, like crisis management as the resolution of tensions between negative and positive ten-

dencies (psycho-social theory, Erikson), want satisfaction (psychoanalytical theories) with goals

such as freedom, justice and the dignity of man (humanistic psychology), tendencies to create a

balance (systemic directions) between the individual and the world, between different structural

elements (schemes) and between scheme and structure (Piaget), targets and the solution of prob-

lems (Case), actions of the greatest possible variety and structural complexity (contextual theo-

ries), self-control of the individual (Thomae, Lerner, et al.), but also interaction with other peo-

ple and role changes (Bronfenbrenner) (Cf. Flammer 1999).

The psychological theories supply arguments for cross-curricular teaching. In particular, the

cross-curricular components alien-ness and co-operation seem to promote student orientation:

Integrating alien aspects promises variety, and specific aspects gain in importance (Cf. applied

maths instruction). The new contents and methods create new, possibly even more complex op-

portunities for action. Through variety one can expect that more attention will be given to the

new information, in that students are given more opportunities to connect with existing struc-

tures, but also that individual processes can be stimulated more often. In addition, integrating

other subjects makes role changes possible, respectively the interaction with people representing

different roles etc. .

The theoretical discussion, touched upon here, can be understood in the way that cross-curricular

teaching may be a practical approach to student orientation. However, it must not be overlooked

that quite general terms clash here. The complexity of cross-curricular teaching becomes obvi-

ous in the model, and Meyer even notes (Meyer 1994, p.110) :

“General targets such as openness, student orientation etc. sound demanding and interest-

ing, however, their disadvantage is that they are only able to be grasped precisely after

awkward work, requiring many additional decisions and still remaining controversial.”

Some of the following arguments for cross-curricular teaching also concern student orientation

and make it more precise. They consider, in particular, also affective and social aspects.

TMME, vol6, Supplement1, p. 35 3.2.2 Cross-curricular Teaching as a Field for Holistic Learning

Next to student orientation, holistic learning (which is inseparable from student orientation and

makes it more precise) is an argument that is often put forward in favour of cross-curricular

teaching (Tenorth 1997). Holistic learning can be characterized very graphically with Pesta-

lozzi’s famous triad “Learning with head, heart and hand” and considers emotional learning and

learning by doing important components of human development.

Holistic learning is characteristic of humanistic education, which, among others, can be justified

by humanistic psychology but also by the structural genetic tradition (Baldwin, Piaget). In hu-

manistic psychology, learning is understood as a“deeply human, personal, affective experience”,

… as a “personal discovery of meaning”, because we “have feelings or emotions if events are

important to us” (Fatzer 1998, p.66, 67). Humanistic psychology puts emphasis on the self-

concept as a factor in human development. The decisive condition for development is the accep-

tance of one’s self, respectively the feeling of being accepted (Flammer 1999). Accordingly, the

basis of humanistic psychology is the conception of man as an entity of spirit, body and soul.

Education is to promote this entity (Fatzer 1998, p.18). Its pedagogical components are given as:

respect and good will, non-prescriptiveness, truth and warmth (Rogers, cf. Flammer 1999), an

emphasis on intrinsic learning (Maslow), the “development of the self” through consciousness of

one’s own motives, abstract thinking and conscious action (Weinstein, Fatzer 1998) as well as a

consideration for the student’s need for identity, belonging and self-determination (“Affective

Education Program” according to Newberg).

In its implementation in school, this leads to an instruction that is oriented to experience and

therefore cross-curricular (Cf. Fatzer 1998). Humanistic pedagogues in particular, like Goodlad,

see a strong tendency to prescriptiveness in the canon of subjects, that contradicts the develop-

ment of the self. In the following quotation from a paper, delivered at the fourth national confer-

ence on humanistic education 1977 in Georgia, Goodlad distinguishes between a hard and a soft

(humanistic) education (Goodlad in Fatzer 1998, p. 49):

“Der harte rigide Stil umfaßt klar definierte und voneinander abgetrennte Fächer oder

Disziplinen….

Für den weichen und zarten Stil …. Ein breites Angebot an Lernzielen mit einem

Hauptgewicht auf Verstehen und Wahrnehmen des eigenen Selbst ist typisch. Eine

Beckmann

Tendenz zur Integration verschiedener Fächer oder zum sog. ´integrativen Tag´ ist ein

weiteres Merkmal dieses Ansatzes….“

Another approach to holistic teaching (through interactive-ness) originates from the structural

genetic tradition. The central concepts, e.g. in Baldwin and Piaget, are scheme and structure,

assimilation and accommodation as well as the circular reactions. Assimilation is the “adapta-

tion by the subject of an environment to the possibilities for action, respectively to the epistemo-

logical conditions of the subject”; accommodation conversely means the “adaptation of the ac-

tional and epistemological possibilities to the demands of the situation” (Flammer 1999, p.117f).

Schemata are the given or acquired epistemic possibilities that are assimilated to the environ-

ment, respectively that accommodate themselves. The (cognitive) structures are the organized

connections of the schemata. The circular reactions describe the close connection and the dy-

namic exchange between assimilation and accommodation. According to Flammer, they are the

essential core of developmental processes (Flammer 1999): after the scheme has been activated,

circular reactions keep it active, they activate a (positive) scheme for another opportunity and

vary it spontaneously. This theory gained special importance for schools through Aebli, whose

two volume work has the significant title “Denken: das Ordnen des Tuns”(Aebli 1980, 1981)

and who speaks for interactive teaching (Cf. Gudjons 1997) with his thesis “Thinking issues

from action” (Aebli 1980, p.26).

On the basis of the theories dealt with, cross-curricular, respectively subject-integrative teaching

seems to be able to comply well with the demand for a holistic approach, in that it shows signs

of an orientation towards experience and action (Cf. Gudjons 1997, Beckmann 1999) as well as

meaning, and therefore (possibly) offers more opportunities for assimilation and accommodation

processes or circular reactions. Cross-curricular teaching in particular allows holistic teaching

methods, such as “group work, partner work, project work, story telling, various forms of scenic

representation, freeze frames, role play, simulation games, experiments, exploration” etc. (Jank,

Meyer 1994, p.356, cf. also Fatzer 1998 and following paragraphs).

3.2.3 Cross-curricular Teaching as a Particular Opportunity for Motivation

Motivation is essential for the understanding of human learning. Motivation influences and

guides a particular behaviour. To be more precise, motivation means “the active concentration of

TMME, vol6, Supplement1, p. 37 one’s present conduct on a positively valued target. This alignment involves different processes

with regard to behaviour and experience” (Rheinberg, 1997, p.14).

Different theories of motivation name different motivational variables, e.g. expectation and re-

ward, and in particular excitement. A condition of optimal excitement is highly conducive to

learning. According to Berlyne, newness, meaningfulness, complexity and the surprise effect of

a stimulus affect increased excitement (Lefrancois 1994, p.192). Cross-curricular teaching then

seems to be interesting from a motivational-psychological point of view (cf. also 3.2.1).

3.2.4 Cross-curricular Teaching as Field for a New Way of Thinking

For the last few years, there often has been a demand for a “new way of thinking”- e.g. in poli-

tics and the sciences – that is no longer linear but multiple and linked; a way of thinking that

allows us to grasp the complexity of our environment and our day to day lives. Reality seems

linked and gives feedbacks: Every change has multiple causes and effects (Flammer 1999).

The “multiply linked global problems of mankind do not only call for technical, scientific or

sociological, philosophical knowledge, but more than ever for holistic thinking, that knows how

to integrate the different subject perspectives” (Mainzer 1993, p. 41). Mainzer makes this idea

more concrete by a cross-curricular question (Mainzer 1993. p.46):

“How can the ecological problems of today’s industrial society be solved with the mod-

ern instruments of the market economy and of technology, while observing an environ-

mental ethic?”

There is a demand for a form of thinking that, with every action or with every phenomenon,

takes into consideration all (or as many as possible) causes and consequences respectively link-

ages.

The demand for linked thinking speaks clearly in favour of cross-curricular teaching, since it is

there that multiplicity and linking is achieved by integrating, respectively mixing alien contents,

methods, ways of thinking etc. . It allows a view of the whole, of the “phenomenon”as such.

Wagenschein characterizes it like this (Wagenschein 1998, p.135, 136):

“Understanding implies: standing by the phenomena” …”and in such a way that we see

them as counterparts and allow them to have an effect on us, yet without prejudice and

without intervention, just impartially, not fixed to a particular aspect.”

Beckmann Wagenschein reminds us here of the lack of links between the school subjects, and that each

subject only ever teaches a single aspect or part of the total reality. He also wants to remind us

of the common educational task of all subjects.

It should not be overlooked, however, that the demand for linked thinking also speaks in favour

of subject teaching, because the ability to grasp the links is in many cases only possible with

thorough subject knowledge.

3.2.5 Cross-curricular Teaching as an Opportunity to Reflect on Subject-

specific Methods

The demand for linked thinking goes together with the demand to organize learning processes in

a non-linear way. It does not do to teach or to learn subject-specific methods. They should also

be turned into topics and students should reflect on them. This can be achieved through a diver-

sified methodological access to a topic, comparison and making it thematic. That way, learning

becomes more co-operative and student-oriented, and it is less direct. In secondary II it may lead

to a form of science propaedeutics. It is obvious, that this in particular is an argument for cross-

curricular teaching - especially in secondary II (Acker 1977).

Tenorth approaches the argument from a different angle, in that he understands the subject

canon as “a structure of certainty” (Tenorth 1997, p.20). In transferring Popper’s concept of fal-

sification, the canon does not become comprehensible by what it is but by what it excludes, “be-

cause the canon is a kind of mark defining for those, who are allowed to learn it, what they are

not allowed to learn” (Tenorth 1997, p. 20). According to Tenorth, cross-curricular instruction

becomes meaningful exactly because it turns the exclusion into the topic.

“At this point I dare say what cross-curricular learning is, namely that it turns the certain-

ties of school knowledge itself - which are usually and quite self-evidently transformed -

into the topic of learning at school.” (Tenorth 1997, p.20).

3.2.6 Cross-curricular Teaching as a “Counterpart” to Specialization

The reflection about subject-specific methods, made possible through the transgression of sub-

jects, also makes it possible to create an awareness of the limits of specialization. This is inter-

esting insofar as, historically, science has always been geared towards specialization and with

good reason, as the following excerpt about the historical view of science shows:

TMME, vol6, Supplement1, p. 39

The origins of the scientific disciplines, including those of the school subjects, go back to

Platonic-Pythagorean philosophy. In the ancient faculties, one distinguished between two

groups of subjects: those subjects that worked out the “true laws and relative proportions

of the cosmos” included arithmetic, geometry, astronomy and music and the subjects of

“logical thought, judgement and argument” which included logic, dialectics and rhetoric

(Mainzer 1993, p.19).

The discussion on the concept of science was particularly influential in the division of

the subjects. According to Aristotle, only purely theoretical subjects counted, such as

mathematics, physics, inclusive of all disciplines that described nature, and metaphysics.

The concept of science in modern times, however, deviates from the prerequisite that it

has to be theoretical and acknowledges, e.g. experimental physics with its mathematical

and technological methods is recognized as a science. Kant had a particular influence on

the division of disciplines in Humboldt’s university. His division of disciplines was

based on the terms “usefulness” and “truth” with philosophy marked off. The useful fac-

ulties were theology, jurisprudence and medicine (upper faculties), while it was to be the

task of philosophy (lower faculties) to present the truth for the use of and also the control

of the upper faculties (Kant 1968). Different paths to the recognition of truth led Kant,

additionally, to divide the philosophical faculty in the fields of “historical recognition”

(history, humanism with the natural sciences, scholarly acquisition of languages, descrip-

tion of the earth) and the fields of “pure logic” (pure mathematics, philosophy, the meta-

physics of nature and of manners) (Kant 1968, Mainzer 1993, p.21). Kant’s definition of

the philosophical faculty is an important example of trans-disciplinariness (cross-

curricular teaching) in history, that fundamentally characterized the spirit of Humboldt’s

university.

In the 19th century, the philosophical faculty could not be maintained. Developments in

the subjects, in particular with regard to experimental methods, led to a delineation of the

natural sciences (and mathematics also), however, with philosophical-theoretical accom-

paniment. While Hegel’s philosophical approach did not consider the natural sciences to

be scientific, mathematics is according to Compte’s positivistic view of the sciences, the

highest scientific discipline, followed by physics, chemistry, biology and sociology

(Mainzer 1993). Compte’s classification of the disciplines according to their positivistic

Beckmann

degree excludes all forms of inter-disciplinary activity, as this would contradict the natu-

ral limits of the subjects, that the categories separate) Vonéche 1993).

Progress in the other faculties led to the establishment of the humanities with their own

methods. Dilthey’s concept of the humanities is a very extended one and comprises all

disciplines whose object is “the historical-social reality”, including language, art, relig-

ion, as well as education, society, law, economy and technology. This division failed to

survive, however, as the rapid progress of the individual disciplines with their own meth-

ods and contents, but also the social development, made discrete fields of education nec-

essary. The industrialization in the 19th

century, in particular, required a discrete training

of engineers. A characteristic of the universities, that originated from the dynamic devel-

opment of the sciences, was the still existing “pluri-disciplinariness” or “multi-

disciplinariness” with many disciplines existing side by side, where at best a linear ex-

change of individual information is practised (Vonéche 1993 according to Piaget, p.116,

Mainer 1993, p.23).

In more recent times, pragmatism has replaced the search for the limits of disciplines. Multi-

disciplinariness is discussed all over the world on the background of general environmental and

humanitarian problems. On the one hand, multi-disciplinariness enables specializations that may

lead to particular scientific findings and developments, necessary for the solution of problems.

On the other hand, the global problems of mankind require linked thinking (Cf. 3.2.4). The need

for cross-curricular questions and their joint solution is accounted for in the scientific commu-

nity through the promotion of inter-disciplinary projects, the establishment of inter-disciplinary

centres (Arber 1993), but also through the endeavour of imparting cross-curricular thinking to

students in their training. Inter-disciplinariness in research is, however, today of a rather reactive

nature, in that it is carried out primarily in those areas where (global) problems have to be

solved, that individual sciences can no longer cope with.

3.2.7 Cross-curricular Teaching as an (additional) Opportunity for Learning

Important Basic Mental Techniques

This objective is not to be seen independent of other objectives – because every objective re-

quires certain basic mental techniques – and equally concerns the objectives of maths teaching

(Zech 1998). Since the objectives are not subject specific, it is logical to try and achieve them

TMME, vol6, Supplement1, p. 41 through cross-curricular questions (Cf. annotations to individual points). However, it remains an

open question at this point, in how far cross-curricular teaching can really achieve these goals

better than pure subject teaching. Because of its structure, cross-curricular teaching is possibly

more concerned with the higher levels of basic mental techniques. (The following collection was

taken from Zech 1998, p. 56ff):

• Comparing (Cf. 3.2.5)

Grasping differences and common features:

Cross-curricular/subject-integrative teaching promises to compare the terms and methods

of different subjects

• Ordering (Cf. 3.2.9)

Collecting, respectively, establishing a rising or falling progression with regard to one or

more criteria:

Cross-curricular/subject-integrative teaching promises, e.g., to order terminological ex-

periences from different fields.

• Abstracting

Collecting essential and neglecting inessential characteristics for a concrete objective or

question:

Cross-curricular/subject-integrative teaching promises to support this process through the

insight that the approaches of different subjects or a cross-curricular problem cause dif-

ferent processes of abstraction.

• Generalising

Collecting common, and at the same time, essential characteristics and forming classes:

Cross-curricular/subject-intensive teaching promises to support this process in that the

co-operation of several subjects validates new fields and shows new applications.

• Classifying

Assigning an object to a class or relating classes:

Cross-curricular/subject-integrative teaching promises a variety of areas where classifica-

tion can be applied.

• Substantiating respectively specializing

Transferring and applying the common to the special features and the individual features:

Cross-curricular/subject integrative teaching enables a review of one’s own subject, such

as in complete orientation.

• Formalizing

In a general sense, as an encoding of information, but also as mathematising:

Beckmann

Cross-curricular/subject-integrative teaching offers a lot of material through situational

applications.

• Finding analogies

Establishing analogies between different areas of phenomena:

Cross-curricular/subject-integrative teaching seems to be predestined for this, since it of-

fers various areas of phenomena through the participation of different subjects.

3.2.8 Cross-curricular Teaching as a Field in which to Experience the Social

Reality of Science

While student orientation and the demand for holistic learning are essential arguments for cross-

curricular teaching in primary and secondary I (and II), the demand for teaching the social real-

ity of science mainly concerns instruction in secondary II. Cross-curricular teaching offers itself

for this, because it may symbolize – for a number of mankind’s problems – a counter-weight

against specialization and it contains the possibility to thematize subject specific methods. This

was already addressed in 3.2.5 and 3.2.6. To be more precise: It may be an objective of cross-

curricular teaching to clarify the relationship between subjects - typical of research. (Huber

1997, p.65):

• Complementary: One view or experience complements the other

• Concentric: Several views address a common subject or problem area

• Contrastive or dialogical: one view or experience contradicts another, makes it relative; it

is a matter of mutual understanding and translating

• Reflexive: with the help of other views, consciously taken, one’s own or that of one’s

subject is reflected upon.

The aspects contain, at the same time, the social component of scientific work. It may be an ob-

jective of cross-curricular teaching to make students experience this, e.g. if general questions in

project work are worked in social interaction (work groups, study groups, planning, communi-

cating, discussing the results etc. (Cf. also 3.2.10).

3.2.9 Cross-curricular Teaching as an Aid in Integrating and Structuring

Learning

TMME, vol6, Supplement1, p. 43 As a rule, the basis of today’s education is the instruction in individual subjects. Students learn

in a subject-specific way. However, school also has the duty to care for the children’s and the

young adults’ integration in the professional, social and cultural reality. In order to unlock life’s

reality, subject elements have to be connected, because reality is holistic (Cf. Wagenschein’s

quotation in 3.2.4). Cross-curricular teaching offers the opportunity to integrate and structure

these learning experiences (Landesinstitut für Schule und Weiterbildung , Soest 1995; cf. also

the argument for linked thinking in 3.2.4).

3.2.10 Cross-curricular Teaching as a Field for the Improved Practice of Gen-

eral Competences

Competences are personal conditions, knowledge and experiences that determine our behaviour

and our actions. Competences are not directly visible but can only be deduced from certain repe-

titions in a person’s behaviour (Flammer 1999). In the following model, four competences are

distinguished (the first three, formulated by Habermas, are complemented by a fourth, formu-

lated by Kreft, Fritzsche 1999, p. 38 ff):

• a cognitive competence, that concerns the recognition of natural laws, if – then-

relationships (dealing with objects)

• an interactive competence, that concerns inter-personal actions and the standards that

they are judged by (dealing with subjects)

• a language competence, that concerns the rules of how words, sentences and texts are

formed, related to reality and understood

• an aesthetic competence, that concerns man’s capability to experience something and to

symbolically express his/her subjectivity.

Social developments can be shown to have a clear influence on the field of human competences.

The enormous progress in subjects, such as the technological developments of the last century

and the growing participation in it by the population, e.g. in the use of the automobile, the inter-

net and the mobile phone, all of these point to an imbalance in the distribution of competences

among people. Various US-American studies, indeed, show a predominance of the cognitive

competence (Fritzsche 1999). However, a purely cognitive alignment is socially unacceptable.

School also has to help develop mature personalities with social responsibility in a democratic

society, which requires taking into account all competences. In addition, a person’s cognitive

Beckmann competence gains further in personal and social importance through the ability to co-operate and

to communicate.

With regard to cross-curricular instruction, the competence model leads to the following consid-

erations:

There is no unambiguous allocation of subjects and competences. Promoting compe-

tences must be seen as a joint task of all subjects, with each subject making its own con-

tribution to the field of competences. This can only be achieved through the respective

different aspects (alien aspects). Cross-curricular teaching allows us to approach this tar-

get jointly. The promotion of individual competences may be strengthened by this and

may be achieved in a more target-oriented way.

On this background, the competence model was variously used to criticize the existing

curriculum. One consequence of the criticism was, e.g. the replacement of the subjects by

“fields of experience” in the Bielefeld laboratory school.

Instruction in subjects can be particularly assigned to the cognitive competence. In that

way, the competence model confirms the importance of a division in subjects, respec-

tively, of working in subjects.

Instruction has to create situations, in which competences can be acquired. With regard to the

non-cognitive competences, a number of methodological suggestions have been made, such as

product-oriented, interactive, student-centred and, in particular, holistic teaching with its various

methods (Cf. 3.2.2). These methods can also be used in subject-specific teaching. However,

cross-curricular teaching, due to its variety of contents and its methodological possibilities as

well as forms of co-operation, seems particularly suited for the use of such methods and there-

fore predestined to promote competences generally. Huber gives the following reasons (Huber

1997, p. 24):

“If we imagine cross-curricular instruction, in which students of different specialization

meet and have to impart their knowledge mutually, in which they mutually answer ques-

tions as experts or have to co-ordinate their various capabilities in projects, then we have

got a situation, in which communicative, co-operative competences, messaging and self-

regulatory competences have not only been preached but made situational and could ac-

tually be experienced.”

TMME, vol6, Supplement1, p. 45 3.2.11 Cross-curricular Teaching as an Opportunity to Develop Ways to Deal

with Heterogeneity

“Starting from the assumption that ever increasing differences in the learning conditions,

the educational biographies, the cultural backgrounds, or even in the ages of the students,

their professional experiences or the experiences they have made during their schooling,

come to bear on our schools … the task of dealing with these differences is a task that is

only confronting us in its totality in the near future. Here, cross-curricular teaching with

its special possibilities to access the problems methodologically and from a content point

of view, offers an enormous opportunity” (Huber 1997, p.23).

The problem Huber addresses in his quotation, does not only concern the students’ school life

but the total social life. In particular, a society with great social variety requires a well trained

way to deal with heterogeneity. Dealing with heterogeneity becomes problematic if it is not seen

as an enrichment but as a disturbance, if there is a clear division between the national and the

alien culture. From a sociological point of view, one’s own culture and the alien culture are in-

terrelated but changeable and most of all relative terms. The concept of the alien is always con-

nected with social exclusion. “What is considered ‘alien’ depends on the image the respective

society has of itself and of its understanding of who makes up the ‘we’ group, of its anxieties, its

ideals and its patterns of perception (Eickelpasch 1199, p. 106). A change in the patterns of per-

ception may lead to a situation, in which what is alien is no longer perceived as alien and is no

longer excluded. To achieve this, should be an educational concern.

In the cross-curricular model, alienness is an essential element but the relativity of the term is

also evident here. What, from a subject’s point of view, is an alien aspect, becomes an idiosyn-

cratic aspect, if viewed from the co-operating subject (Cf. model). One can safely assume that

alien-ness, such as a German language teacher’s fear of mathematical formalism, forestalls or

inhibits co-operation. In the cross-curricular model, alien-ness is, however, explicitly called an

enrichment. Without alien-ness there would be no motivation for cross-curricular teaching.

Cross-curricular teaching then offers the special opportunity to experience alien-ness as an ex-

plicit enrichment (Cf. also to 3.2.8).

Beckmann 3.2.12 Cross-curricular Teaching as a Contribution to General Education

General education is a very comprehensive term, characterized by varying emphases in different

educational theories. In classic educational theories, there were three criteria (which, in fact,

clearly favour cross-curricular teaching (Jank, Meyer 1994, p 175 f.):

• General education is education for all without regard to provenance, parentage, social

class or fortune.

• General education is comprehensive education, i.e. an education that promotes an indi-

vidual in the greatest possible way – not only in special fields or with respect to his/her

later profession – but in all his/her capabilities.

• General education is education “in a general sense”, i.e. in the comprehensive context of

the world around us.

In the present discussion about education, two directions can be distinguished: One direction

defines general education via a canon of subjects, contents, knowledge, competences etc. . The

other direction identifies methods and competences that man needs to live, survive and work

(Jank, Meyer 1994). In contrast to these, there is Klafki, who rejects the formulation of canons

for general education. For him, the central issue is the solution of present and future key prob-

lems (Klafki in Jank, Meyer 1994, p.177):

• ”My thesis is as follows: Education, respectively, general education … means to have

gained a consciousness of the central problems of our common present and predicable

future, transmitted through history, an insight into our collective responsibility for such

problems and the willingness to face them and take part in their solution…”

In paragraph 3.2.13, the contribution, cross-curricular teaching can make to general education,

will be discussed.

3.2.13 Cross-curricular Teaching as a Special Opportunity to Deal with Topi-

cal Issues

In a world, growing together more closely all the time, catastrophes or extraordinary events are

becoming more and more likely. This should not be a no-no for schools – on the contrary, they

should address topical issues (This also concerns the general education aspect in 3.2.12). Such

issues are rarely subject-specific. As an example, an accident in a power station is related to

TMME, vol6, Supplement1, p. 47 physical (cause), chemical, biological (effect), but also political, economic, geographical, socio-

logical aspects, and, last but not least, to mathematical aspects. Cross-curricular teaching offers

the opportunity to comprehensively understand the topical event. This implies, however, that

cross-curricular teaching can be effected spontaneously. The organizational prerequisites would

have to be clarified beforehand. As shown in the magnet and star mode, the colleagues from

different departments would be able to recognize the relevance of it for their departments instan-

taneously and be able to make their contributions.

3.2.14 Cross-curricular Teaching as a Special Opportunity to Disclose the Im-

portance of Interdisciplinary Co-operation in the Solution of Problems

The importance of interdisciplinary learning and linked thinking for the solution of the global

problems of mankind has already been addressed at various points. However, interdisciplinary

learning can also be of importance for the solution of technical questions. Zeidler (1998) shows

this very impressively in his contribution “The fascination of the mutual effects between

mathematics and the natural sciences” (my translation). Mathematics, e.g., allows us to integrate

the myriad of natural phenomena in a few basic equations; and the collaboration between

mathematical and physical research programmes can lead to fascinating insights: The realisation

“Energy equals Curvature”, f. e., resulted from physical research into the energy concept and

mathematical research into the curvature of geometrical forms. However, in order to bring to-

gether such research areas, an intensive dialogue is necessary, for which the foundations can be

laid in school. Zeidler concludes (Zeidler 1998, p. 335):

“It is an important task for the future to train young people in school and at universities

who are prepared to and able to enter into a dialogue with different scientific disci-

plines.”

Cross-curricular teaching can be seen as an opportunity to highlight its importance at school

level. One objective of cross-curricular teaching may be to make students aware that “the solu-

tion of problems is facilitated, if they are structured along subject aspects with the aim of bring-

ing together the results in a cross-curricular way” (Landesinstitut für Schule und Weiterbildung,

Soest 1995).

Beckmann 3.2.15 Cross-curricular Teaching as an Opportunity to Solidify Subject

Knowledge

The basis of this objective is the idea that, through the cross-curricular problem, questions can

be approached that otherwise remain undisclosed, respectively cannot be dealt with in the same

complexity. Thus, in a German language class, topics of comprehensive orientation are better

suited to a discussion than purely literary or linguistic questions. For maths teaching this has to

be seen in the context of the rationale for applied mathematics and the importance of mathemati-

cal model formation. It is obvious, however, that cross-curricular teaching that is designed to

delve especially deeply into a subject, has to have long subject specific phases. In project orien-

tation, this could be done in work groups. However, this objective is to be examined separately.

For maths teaching, the following applies:

“The concentration on situational aspects, that can be treated mathematically, … may

collide with the interest in other dimensions of the problem” (Jablonka 1999, p. 65).

With regard to a specific subject, this additionally results in the following objectives:

3.2.16 Cross-curricular Teaching as an Opportunity to Experience the Par-

ticular Importance of a Given Subject

According to the cross-curricular/subject-integrative model, there are two types of subject orien-

tation. With regard to the argument in question, the x-orientation is of particular interest. In x-

orientation subject X is the starting point. In co-operation with subject F, X is used (and possibly

worked out) to introduce, work out, explain, describe connections in F, to create objects in F or

to structure F. On the surface, it is about teaching content from F. However, this content is de-

veloped or deduced with the help of X. That way, the importance of X in contexts outside X is

made clear. Its importance is enhanced in the co-operation with various very different subjects,

respectively in connection with topics of comprehensive orientation, that generate personal con-

cernment.

TMME, vol6, Supplement1, p. 49 3.2.17 Cross-curricular Teaching as a Special Opportunity to Tackle the

Problems of a Particular Subject

With the publication of the results of the TIMSS-study (Third International Mathematics and

Science Study) and as a consequence of the PISA-study, German maths and language instruction

was exposed to public criticism. A detailed analysis has shown, however, that the criticism was

only relative in some points (Cf. for maths Blum Neubrand 1998, Wiegand 2000, Weigand

1997); however, certain deficits have to be admitted:

“The deficits in the achievement of German high school student compared with students

from countries with higher student achievement are particularly obvious in tasks that test

mathematical-natural science understanding, demand a flexible application of learned

material or offer an unusual problem constellation” (Baptist 1998, p. 5).

The weaknesses show up

• if a flexible connection of several subject areas is required, i.e. of geometrical tasks to

calculations,

• if several steps have to be combined to solve a task,

• if different aspects of a topic are addressed simultaneously,

• if the use of unaccustomed material is required, i.e. terms not used in the accustomed

contexts,

• if the construction of complex models is expected” (Neubrand, Neubrand, Sibberns

1998, p.26 in Wiegand 2000, p. 95).

Organisational and methodological consequences have been drawn from these results, such

as more frequent use of internal differentiation, more frequent change of methods, lower

teaching loads for teachers (Japanese teachers only teach 16 instead of 25 lessons a week)

etc. (Weigand 1997). Ulm formulated 15 points, including varying the formulation of the

tasks, open questions, but also securing basic knowledge, cumulative learning, setting tasks

applicable to meaningful contexts and tasks requiring subject-integrative work (Ulm 2001).

4 The Implementation

Beckmann The abundance of objectives and arguments for cross-curricular instruction may create the im-

pression, that teaching the core of a specific subject is meaningless. This impression is problem-

atic and wrong. It has already been referred to in some places.

On the one hand, the objectives mentioned do not at all comprise all teaching objectives. On the

other hand, they only make a limited and quite general statement about the actual value they

have for the subject concerned. If, how and to what degree the advantages are achieved, must be

seen in each individual case.

In the implementation of cross-curricular/subject-integrative teaching in school, the model can

be used as a basis for decision making. This assumes, however, a comprehensive survey of the

subjects involved and their connection, as well as a survey of the objectives, that can be reached

or that are aimed at in the co-operation (see section 3).

Thus, in parallel work and parallel orientation, it should be guaranteed that all students in the

group are actually attending instruction in the parallel subjects. This may be difficult, in particu-

lar at the secondary level where specialization is the norm in Germany. A suggested solution is

the “institutionalized coupling model” in which it is mandatory for the students to choose a cer-

tain subject combination (Landesinstitut für Schule und Weiterbildung, Soest 1999).

Additionally, freedom in the choice of methods must be possible, which in project orientation

allows the use of the respective repertoire of methods, including group work, self-study and pos-

sibly extra-curricular events (excursions, study days, performances of plays), or which, in the

reflection on subject-specific methods, chooses alien methods that are really applicable (experi-

ments taken from physics in maths instruction). It becomes clear that there will be a greater de-

mand for special classrooms allowing spontaneous use, for special rooms for subject- integrative

teaching, respectively for extra-curricular days.

“Principally, one must ask the question, in how far the … existing principle of allocating

rooms according to the learner groups must be reconsidered, if and when it makes sense,

or is even indispensable in cross-curricular teaching, that some rooms should be espe-

cially equipped for subject- or project-oriented teaching. This could make the use of ex-

isting aids easier and allow the creation of a collection. In addition, it ought to be consid-

TMME, vol6, Supplement1, p. 51

ered whether rooms for self-study are needed.” (Landesinstitut für Schule und Weiter-

bildung, Soest 1995)

In addition, freedom of planning must be guaranteed by all stakeholders, so that instruction can

be spontaneous (example-orientation at level 1) or prepared long beforehand (course-oriented at

level 4).

Long term planning is characterized by the fact, that cross-curricular/subject-

integrative teaching is preceded by a phase filled with comprehensive theoretical work. This

basis can lead to the arrangement that cross-curricular teaching is part of the curriculum or part

of the school’s program. In that sense, the planners need not be identical with the co-operating

teachers that implement it later on, however a design science approach is warranted to ensure

planning, communication, implementation and research in multiple phases. In a mode of replac-

ing subjects, the long term planning can refer to the development of a focus for the stu-

dents’entire schooling, that is constantly renewed, taken up by other subjects and dealt with in

greater depth. Certain periods at various levels could be allocated to cross-curricular teaching

(Landesinstitut für Schule und Weiterbildung, Soest 1995). In a subject-complementary form,

the long term planning may refer to the development of new integrative subjects, such as the

subject natural science. Long term planning implies a high degree of subject competence, but

also an informed overview of all school levels. Long term planning, that is integrated into the

curriculum, has the advantage that a new form of period, room and staff planning can be intro-

duced for the time that periods are adjourned, which also allows the freedom to try out new

methods. However, this pre-supposes, that all of these subjects are co-ordinated in the curricu-

lum. I argue that the subject complementary form avoids this problem and can be largely inte-

grated into the given arrangement.

In medium-term planning, which can of course be based on curricular suggestions, the co-

operation is usually pre-planned before the start of the school year and only concerns part of the

staff. This requires consideration in the timetable, where team teaching or possibilities for sub-

stituting during the respective time span, play a role. To increase flexibility, the occasional use

of double periods, marginal periods and successive periods for co-operating teachers, respec-

tively free periods are suggested (i.e. Landesinstitut für Schule und Weiterbildung, Soest 1995).

Beckmann It becomes clear that cross-curricular/subject-integrative teaching is easier to arrange through

long-term, central planning than in medium-term planning. It is medium-term or even short-term

planning, however, that makes creativity and new concepts possible, and it also allows teachers

to respond to topical events. Mid-range planning is also desirable. Here, the introduction of a

reservoir of marginal periods in the timetable may help, which can be used by all subject teach-

ers spontaneously and temporarily for cross-curricular teaching, and which can replace regular

teaching for a time.

To conclude, the problem of performance measurement, which is a special topic in project-

orientation, will be addressed briefly. In addition to classroom tests, reports, work progress re-

ports, presentations or papers are suggested (see Landesinstitut für Schule und Weiterbildung,

Soest 1995).

5 Final Remarks This paper shows that although there is no single type of cross-curricular/subject-integrative

instruction with a single set of objectives, there are techniques to make cross-curricular/subject-

integrative teaching multi-layered, differentiated and highly variable. The conceptual model, that

was developed, reflects this complexity.

The four forms of co-operation and their respective three or two co-operative approaches,

the three contact forms, the four interest orientations (disregarding a differentiation be-

tween orientation towards common and alien features) and the two respectively three

forms of interest orientation already result – purely mathematically, without considering

subject aspects and topics - in 200 different possibilities!

In the implementation of cross-curricular teaching, this complexity must be considered, in that

the layout takes into consideration decisions made on the background of the model. If and in

what form cross-curricular teaching seems suitable for the classroom, has to be examined on a

case by case basis. It ought to be considered here, that cross-curricular teaching should always

be aimed at enrichment. The model, presented in 2, offers a basis for decisions and evaluations

TMME, vol6, Supplement1, p. 53 with the background and framework of the objectives in 3, in that it presents the respective com-

ponents of the model that are applicable to individual designs.10

Beckmann, A. (2003). Cross-curricular math lessons, vol 1. Fächerübergreifender Mathematikunterricht

– Ein Modell, Ziele und fachspezifische Diskussion. Hildesheim, Berlin (Franzbecker) – 2003a

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10 The particular value of the model is that it allows a theoretical foundation as didactical and methodological groundwork for the practical implementation of cross-curricular teaching with respect to special subjects. We have done this groundwork for cross-curricular maths teaching. In particular, the possibilities for co-operation with sub-jects such as Physics, German and Information Technology were examined (Beckmann 2003 a, b, c, d).

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Beckmann Cooperation Forms for Cross-Curricular Lessons

Short overview, according to Model Beckmann 200311

Topic and Major Subject – related form (TM - Form)

Aspects (contents, methods,..) from scientific subjects (physics, chemistry, biology..) used in mathemat-ics lessons.

OrganisationCommunication with colleagues teaching scientific subjects,

: Initiative: mathematics teacher,

Colleagues support mathematics teacher

Parallel Topic - related – Form (PT - Form) School year

Mathematics Aspects of the theme con-cerning mathematics learning

Physics Aspects of the theme con-cerning mathematics learning

Chemistry ….

Biology ….

Geography

…..

……

Organisation:Communication and common planning of the school year or parts of it,

Initiative: one or more teachers

Parallel teaching of the same theme and permanent exchange between the teachers during this period

Parallel Planning – Form (PP - Form) Possible example

Mathematics – physics – chemistry – biology Introduction – approaching the theme (in common) Mathematics Special aspect of the theme

Physics Special aspect of the theme

Chemistry Special aspect of the theme

Biology Special aspect of the theme

Mathematics and Physics Mathematical modelling of the physical phenaomena Mathematics – Biology Using mathematics argumentation in biology

Physics Deepening of the physical aspects Mathematics

Deepening the mathe-matics aspects

Biology – Chemistry Discussing common aspects of the theme

Mathematics – physics – chemistry – biology Results (in common) and summary

Organisation:Permanent communication and common planning before and during teaching the modul, partly: common teaching according to the needs of the theme

Initiative: one or more teachers

Joint Planning – Form (JP - Form)

Organisation:Team teaching: All subject melt together to one subject!

Initiative: one or more teachers

Possible: project-oriented teaching with subject-oriented project parts.

11 Beckmann, Astrid (2003). Fächerübergreifender Unterricht – Konzept und Begründung, Hildesheim, Berlin (Franzbecker Verlag), www.sciencemath.ph-gmuend.de